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Properties

Label 12-3800e6-1.1-c1e6-0-7
Degree 1212
Conductor 3.011×10213.011\times 10^{21}
Sign 11
Analytic cond. 7.80484×1087.80484\times 10^{8}
Root an. cond. 5.508465.50846
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s − 4·9-s + 3·11-s + 13-s + 14·17-s − 6·19-s + 4·21-s − 12·23-s + 10·27-s + 9·29-s + 5·31-s − 6·33-s + 8·37-s − 2·39-s + 3·41-s − 15·43-s − 4·47-s − 8·49-s − 28·51-s − 13·53-s + 12·57-s + 9·61-s + 8·63-s + 3·67-s + 24·69-s + 19·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s − 4/3·9-s + 0.904·11-s + 0.277·13-s + 3.39·17-s − 1.37·19-s + 0.872·21-s − 2.50·23-s + 1.92·27-s + 1.67·29-s + 0.898·31-s − 1.04·33-s + 1.31·37-s − 0.320·39-s + 0.468·41-s − 2.28·43-s − 0.583·47-s − 8/7·49-s − 3.92·51-s − 1.78·53-s + 1.58·57-s + 1.15·61-s + 1.00·63-s + 0.366·67-s + 2.88·69-s + 2.25·71-s + ⋯

Functional equation

Λ(s)=((218⋅512⋅196)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅512⋅196)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 218⋅512⋅1962^{18} \cdot 5^{12} \cdot 19^{6}
Sign: 11
Analytic conductor: 7.80484×1087.80484\times 10^{8}
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 218⋅512⋅196, ( :[1/2]6), 1)(12,\ 2^{18} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 4.7129416364.712941636
L(12)L(\frac12) ≈\approx 4.7129416364.712941636
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
19 (1+T)6 ( 1 + T )^{6}
good3 1+2T+8T2+14T3+10pT4+50T5+29pT6+50pT7+10p3T8+14p3T9+8p4T10+2p5T11+p6T12 1 + 2 T + 8 T^{2} + 14 T^{3} + 10 p T^{4} + 50 T^{5} + 29 p T^{6} + 50 p T^{7} + 10 p^{3} T^{8} + 14 p^{3} T^{9} + 8 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
7 1+2T+12T2+22T3+118T4+18pT5+779T6+18p2T7+118p2T8+22p3T9+12p4T10+2p5T11+p6T12 1 + 2 T + 12 T^{2} + 22 T^{3} + 118 T^{4} + 18 p T^{5} + 779 T^{6} + 18 p^{2} T^{7} + 118 p^{2} T^{8} + 22 p^{3} T^{9} + 12 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
11 1−3T+26T2−26T3+65T4+749T5−208pT6+749pT7+65p2T8−26p3T9+26p4T10−3p5T11+p6T12 1 - 3 T + 26 T^{2} - 26 T^{3} + 65 T^{4} + 749 T^{5} - 208 p T^{6} + 749 p T^{7} + 65 p^{2} T^{8} - 26 p^{3} T^{9} + 26 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
13 1−T+pT2−50T3+38T4+516T5+1813T6+516pT7+38p2T8−50p3T9+p5T10−p5T11+p6T12 1 - T + p T^{2} - 50 T^{3} + 38 T^{4} + 516 T^{5} + 1813 T^{6} + 516 p T^{7} + 38 p^{2} T^{8} - 50 p^{3} T^{9} + p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12}
17 1−14T+124T2−848T3+4842T4−24018T5+105929T6−24018pT7+4842p2T8−848p3T9+124p4T10−14p5T11+p6T12 1 - 14 T + 124 T^{2} - 848 T^{3} + 4842 T^{4} - 24018 T^{5} + 105929 T^{6} - 24018 p T^{7} + 4842 p^{2} T^{8} - 848 p^{3} T^{9} + 124 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
23 1+12T+112T2+778T3+4884T4+25856T5+128015T6+25856pT7+4884p2T8+778p3T9+112p4T10+12p5T11+p6T12 1 + 12 T + 112 T^{2} + 778 T^{3} + 4884 T^{4} + 25856 T^{5} + 128015 T^{6} + 25856 p T^{7} + 4884 p^{2} T^{8} + 778 p^{3} T^{9} + 112 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}
29 1−9T+107T2−600T3+5158T4−26640T5+189919T6−26640pT7+5158p2T8−600p3T9+107p4T10−9p5T11+p6T12 1 - 9 T + 107 T^{2} - 600 T^{3} + 5158 T^{4} - 26640 T^{5} + 189919 T^{6} - 26640 p T^{7} + 5158 p^{2} T^{8} - 600 p^{3} T^{9} + 107 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
31 1−5T+50T2−176T3+771T4−5843T5+22284T6−5843pT7+771p2T8−176p3T9+50p4T10−5p5T11+p6T12 1 - 5 T + 50 T^{2} - 176 T^{3} + 771 T^{4} - 5843 T^{5} + 22284 T^{6} - 5843 p T^{7} + 771 p^{2} T^{8} - 176 p^{3} T^{9} + 50 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12}
37 1−8T+132T2−743T3+7797T4−33802T5+309503T6−33802pT7+7797p2T8−743p3T9+132p4T10−8p5T11+p6T12 1 - 8 T + 132 T^{2} - 743 T^{3} + 7797 T^{4} - 33802 T^{5} + 309503 T^{6} - 33802 p T^{7} + 7797 p^{2} T^{8} - 743 p^{3} T^{9} + 132 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
41 1−3T+66T2−146T3+4087T4−5943T5+137612T6−5943pT7+4087p2T8−146p3T9+66p4T10−3p5T11+p6T12 1 - 3 T + 66 T^{2} - 146 T^{3} + 4087 T^{4} - 5943 T^{5} + 137612 T^{6} - 5943 p T^{7} + 4087 p^{2} T^{8} - 146 p^{3} T^{9} + 66 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
43 1+15T+271T2+2530T3+26829T4+185919T5+1470270T6+185919pT7+26829p2T8+2530p3T9+271p4T10+15p5T11+p6T12 1 + 15 T + 271 T^{2} + 2530 T^{3} + 26829 T^{4} + 185919 T^{5} + 1470270 T^{6} + 185919 p T^{7} + 26829 p^{2} T^{8} + 2530 p^{3} T^{9} + 271 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12}
47 1+4T+230T2+749T3+23811T4+1352pT5+1431531T6+1352p2T7+23811p2T8+749p3T9+230p4T10+4p5T11+p6T12 1 + 4 T + 230 T^{2} + 749 T^{3} + 23811 T^{4} + 1352 p T^{5} + 1431531 T^{6} + 1352 p^{2} T^{7} + 23811 p^{2} T^{8} + 749 p^{3} T^{9} + 230 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
53 1+13T+126T2+1554T3+12921T4+108347T5+968797T6+108347pT7+12921p2T8+1554p3T9+126p4T10+13p5T11+p6T12 1 + 13 T + 126 T^{2} + 1554 T^{3} + 12921 T^{4} + 108347 T^{5} + 968797 T^{6} + 108347 p T^{7} + 12921 p^{2} T^{8} + 1554 p^{3} T^{9} + 126 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12}
59 1+161T2+255T3+12605T4+44407T5+744938T6+44407pT7+12605p2T8+255p3T9+161p4T10+p6T12 1 + 161 T^{2} + 255 T^{3} + 12605 T^{4} + 44407 T^{5} + 744938 T^{6} + 44407 p T^{7} + 12605 p^{2} T^{8} + 255 p^{3} T^{9} + 161 p^{4} T^{10} + p^{6} T^{12}
61 1−9T+234T2−2302T3+30205T4−3881pT5+2409968T6−3881p2T7+30205p2T8−2302p3T9+234p4T10−9p5T11+p6T12 1 - 9 T + 234 T^{2} - 2302 T^{3} + 30205 T^{4} - 3881 p T^{5} + 2409968 T^{6} - 3881 p^{2} T^{7} + 30205 p^{2} T^{8} - 2302 p^{3} T^{9} + 234 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
67 1−3T+203T2−820T3+23862T4−100928T5+1840467T6−100928pT7+23862p2T8−820p3T9+203p4T10−3p5T11+p6T12 1 - 3 T + 203 T^{2} - 820 T^{3} + 23862 T^{4} - 100928 T^{5} + 1840467 T^{6} - 100928 p T^{7} + 23862 p^{2} T^{8} - 820 p^{3} T^{9} + 203 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
71 1−19T+351T2−3254T3+31709T4−169223T5+1652142T6−169223pT7+31709p2T8−3254p3T9+351p4T10−19p5T11+p6T12 1 - 19 T + 351 T^{2} - 3254 T^{3} + 31709 T^{4} - 169223 T^{5} + 1652142 T^{6} - 169223 p T^{7} + 31709 p^{2} T^{8} - 3254 p^{3} T^{9} + 351 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12}
73 1+3T+133T2+274T3+18744T4+31092T5+1356111T6+31092pT7+18744p2T8+274p3T9+133p4T10+3p5T11+p6T12 1 + 3 T + 133 T^{2} + 274 T^{3} + 18744 T^{4} + 31092 T^{5} + 1356111 T^{6} + 31092 p T^{7} + 18744 p^{2} T^{8} + 274 p^{3} T^{9} + 133 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
79 1+16T+189T2+1815T3+19499T4+139691T5+1154286T6+139691pT7+19499p2T8+1815p3T9+189p4T10+16p5T11+p6T12 1 + 16 T + 189 T^{2} + 1815 T^{3} + 19499 T^{4} + 139691 T^{5} + 1154286 T^{6} + 139691 p T^{7} + 19499 p^{2} T^{8} + 1815 p^{3} T^{9} + 189 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12}
83 1+31T+824T2+13802T3+205891T4+2327315T5+23897248T6+2327315pT7+205891p2T8+13802p3T9+824p4T10+31p5T11+p6T12 1 + 31 T + 824 T^{2} + 13802 T^{3} + 205891 T^{4} + 2327315 T^{5} + 23897248 T^{6} + 2327315 p T^{7} + 205891 p^{2} T^{8} + 13802 p^{3} T^{9} + 824 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12}
89 1−14T+467T2−5503T3+93933T4−916667T5+10735038T6−916667pT7+93933p2T8−5503p3T9+467p4T10−14p5T11+p6T12 1 - 14 T + 467 T^{2} - 5503 T^{3} + 93933 T^{4} - 916667 T^{5} + 10735038 T^{6} - 916667 p T^{7} + 93933 p^{2} T^{8} - 5503 p^{3} T^{9} + 467 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
97 1−11T+374T2−3862T3+74175T4−650639T5+8888572T6−650639pT7+74175p2T8−3862p3T9+374p4T10−11p5T11+p6T12 1 - 11 T + 374 T^{2} - 3862 T^{3} + 74175 T^{4} - 650639 T^{5} + 8888572 T^{6} - 650639 p T^{7} + 74175 p^{2} T^{8} - 3862 p^{3} T^{9} + 374 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.34622309274093397128652057411, −4.15706055029957661762160157046, −4.02797270679690645626290552090, −3.98407109844095702376812415286, −3.88847013151906623993700021614, −3.71298704342400990336202592685, −3.49013750588492898404309316889, −3.16105278521843829131378944880, −3.15702560461685351166608566606, −2.98833209566256860990705508822, −2.94307679812171951738207131986, −2.92181657668714199430125623287, −2.91657404336742598428573155388, −2.47561651512472468204297282998, −2.05859302237289275087996841947, −1.92945799514007907577466530205, −1.86132197103629459089971158830, −1.77939889325735359663450105121, −1.68321993376562308574504357026, −1.36995978014762498193570884767, −0.965009873455249558456152894215, −0.68198939034702020857083765837, −0.58249917694669068976413874887, −0.55777705409897091411123674292, −0.34772825004131021745878878768, 0.34772825004131021745878878768, 0.55777705409897091411123674292, 0.58249917694669068976413874887, 0.68198939034702020857083765837, 0.965009873455249558456152894215, 1.36995978014762498193570884767, 1.68321993376562308574504357026, 1.77939889325735359663450105121, 1.86132197103629459089971158830, 1.92945799514007907577466530205, 2.05859302237289275087996841947, 2.47561651512472468204297282998, 2.91657404336742598428573155388, 2.92181657668714199430125623287, 2.94307679812171951738207131986, 2.98833209566256860990705508822, 3.15702560461685351166608566606, 3.16105278521843829131378944880, 3.49013750588492898404309316889, 3.71298704342400990336202592685, 3.88847013151906623993700021614, 3.98407109844095702376812415286, 4.02797270679690645626290552090, 4.15706055029957661762160157046, 4.34622309274093397128652057411

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.