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Properties

Label 12-1323e6-1.1-c3e6-0-1
Degree 1212
Conductor 5.362×10185.362\times 10^{18}
Sign 11
Analytic cond. 2.26232×10112.26232\times 10^{11}
Root an. cond. 8.835138.83513
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 15·4-s + 24·5-s + 69·16-s + 42·17-s − 360·20-s + 24·25-s − 312·37-s + 360·41-s + 654·43-s + 1.81e3·47-s − 6·59-s + 141·64-s + 42·67-s − 630·68-s + 1.95e3·79-s + 1.65e3·80-s + 2.89e3·83-s + 1.00e3·85-s + 1.51e3·89-s − 360·100-s − 456·101-s + 2.55e3·109-s − 2.40e3·121-s − 1.23e3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.87·4-s + 2.14·5-s + 1.07·16-s + 0.599·17-s − 4.02·20-s + 0.191·25-s − 1.38·37-s + 1.37·41-s + 2.31·43-s + 5.62·47-s − 0.0132·59-s + 0.275·64-s + 0.0765·67-s − 1.12·68-s + 2.78·79-s + 2.31·80-s + 3.82·83-s + 1.28·85-s + 1.80·89-s − 0.359·100-s − 0.449·101-s + 2.24·109-s − 1.80·121-s − 0.884·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

Λ(s)=((318⋅712)s/2 ΓC(s)6 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((318⋅712)s/2 ΓC(s+3/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 318⋅7123^{18} \cdot 7^{12}
Sign: 11
Analytic conductor: 2.26232×10112.26232\times 10^{11}
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 318⋅712, ( :[3/2]6), 1)(12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 20.2434168920.24341689
L(12)L(\frac12) ≈\approx 20.2434168920.24341689
L(52)L(\frac{5}{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)
bad3 1 1
7 1 1
good2 1+15T2+39p2T4+291p2T6+39p8T8+15p12T10+p18T12 1 + 15 T^{2} + 39 p^{2} T^{4} + 291 p^{2} T^{6} + 39 p^{8} T^{8} + 15 p^{12} T^{10} + p^{18} T^{12}
5 (1−12T+204T2−654pT3+204p3T4−12p6T5+p9T6)2 ( 1 - 12 T + 204 T^{2} - 654 p T^{3} + 204 p^{3} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} )^{2}
11 1+2406T2+2486343T4+1930638516T6+2486343p6T8+2406p12T10+p18T12 1 + 2406 T^{2} + 2486343 T^{4} + 1930638516 T^{6} + 2486343 p^{6} T^{8} + 2406 p^{12} T^{10} + p^{18} T^{12}
13 1+4677T2+15059643T4+218103758p2T6+15059643p6T8+4677p12T10+p18T12 1 + 4677 T^{2} + 15059643 T^{4} + 218103758 p^{2} T^{6} + 15059643 p^{6} T^{8} + 4677 p^{12} T^{10} + p^{18} T^{12}
17 (1−21T+9654T2−298065T3+9654p3T4−21p6T5+p9T6)2 ( 1 - 21 T + 9654 T^{2} - 298065 T^{3} + 9654 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} )^{2}
19 1+10698T2+33088839T4−140201322100T6+33088839p6T8+10698p12T10+p18T12 1 + 10698 T^{2} + 33088839 T^{4} - 140201322100 T^{6} + 33088839 p^{6} T^{8} + 10698 p^{12} T^{10} + p^{18} T^{12}
23 1+50265T2+1230954627T4+18571227575238T6+1230954627p6T8+50265p12T10+p18T12 1 + 50265 T^{2} + 1230954627 T^{4} + 18571227575238 T^{6} + 1230954627 p^{6} T^{8} + 50265 p^{12} T^{10} + p^{18} T^{12}
29 1+3537pT2+5093278491T4+155279545452270T6+5093278491p6T8+3537p13T10+p18T12 1 + 3537 p T^{2} + 5093278491 T^{4} + 155279545452270 T^{6} + 5093278491 p^{6} T^{8} + 3537 p^{13} T^{10} + p^{18} T^{12}
31 1+113865T2+6700832043T4+247428066014534T6+6700832043p6T8+113865p12T10+p18T12 1 + 113865 T^{2} + 6700832043 T^{4} + 247428066014534 T^{6} + 6700832043 p^{6} T^{8} + 113865 p^{12} T^{10} + p^{18} T^{12}
37 (1+156T+147900T2+15097686T3+147900p3T4+156p6T5+p9T6)2 ( 1 + 156 T + 147900 T^{2} + 15097686 T^{3} + 147900 p^{3} T^{4} + 156 p^{6} T^{5} + p^{9} T^{6} )^{2}
41 (1−180T+149496T2−14321250T3+149496p3T4−180p6T5+p9T6)2 ( 1 - 180 T + 149496 T^{2} - 14321250 T^{3} + 149496 p^{3} T^{4} - 180 p^{6} T^{5} + p^{9} T^{6} )^{2}
43 (1−327T+196824T2−36989763T3+196824p3T4−327p6T5+p9T6)2 ( 1 - 327 T + 196824 T^{2} - 36989763 T^{3} + 196824 p^{3} T^{4} - 327 p^{6} T^{5} + p^{9} T^{6} )^{2}
47 (1−906T+390030T2−124576836T3+390030p3T4−906p6T5+p9T6)2 ( 1 - 906 T + 390030 T^{2} - 124576836 T^{3} + 390030 p^{3} T^{4} - 906 p^{6} T^{5} + p^{9} T^{6} )^{2}
53 1+180597T2+71733427059T4+7770811583749470T6+71733427059p6T8+180597p12T10+p18T12 1 + 180597 T^{2} + 71733427059 T^{4} + 7770811583749470 T^{6} + 71733427059 p^{6} T^{8} + 180597 p^{12} T^{10} + p^{18} T^{12}
59 (1+3T+324096T2−59668521T3+324096p3T4+3p6T5+p9T6)2 ( 1 + 3 T + 324096 T^{2} - 59668521 T^{3} + 324096 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} )^{2}
61 1+782898T2+282242036535T4+70824652394396060T6+282242036535p6T8+782898p12T10+p18T12 1 + 782898 T^{2} + 282242036535 T^{4} + 70824652394396060 T^{6} + 282242036535 p^{6} T^{8} + 782898 p^{12} T^{10} + p^{18} T^{12}
67 (1−21T+805641T2−16881406T3+805641p3T4−21p6T5+p9T6)2 ( 1 - 21 T + 805641 T^{2} - 16881406 T^{3} + 805641 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} )^{2}
71 1+718233T2+448945913091T4+180928540927620678T6+448945913091p6T8+718233p12T10+p18T12 1 + 718233 T^{2} + 448945913091 T^{4} + 180928540927620678 T^{6} + 448945913091 p^{6} T^{8} + 718233 p^{12} T^{10} + p^{18} T^{12}
73 1+859254T2+509280578079T4+226900074133018100T6+509280578079p6T8+859254p12T10+p18T12 1 + 859254 T^{2} + 509280578079 T^{4} + 226900074133018100 T^{6} + 509280578079 p^{6} T^{8} + 859254 p^{12} T^{10} + p^{18} T^{12}
79 (1−978T+1121190T2−639733376T3+1121190p3T4−978p6T5+p9T6)2 ( 1 - 978 T + 1121190 T^{2} - 639733376 T^{3} + 1121190 p^{3} T^{4} - 978 p^{6} T^{5} + p^{9} T^{6} )^{2}
83 (1−1446T+1946202T2−1587114996T3+1946202p3T4−1446p6T5+p9T6)2 ( 1 - 1446 T + 1946202 T^{2} - 1587114996 T^{3} + 1946202 p^{3} T^{4} - 1446 p^{6} T^{5} + p^{9} T^{6} )^{2}
89 (1−759T+1423131T2−991050762T3+1423131p3T4−759p6T5+p9T6)2 ( 1 - 759 T + 1423131 T^{2} - 991050762 T^{3} + 1423131 p^{3} T^{4} - 759 p^{6} T^{5} + p^{9} T^{6} )^{2}
97 1+1875102T2+3279229040895T4+3114657711972625988T6+3279229040895p6T8+1875102p12T10+p18T12 1 + 1875102 T^{2} + 3279229040895 T^{4} + 3114657711972625988 T^{6} + 3279229040895 p^{6} T^{8} + 1875102 p^{12} T^{10} + p^{18} 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

−4.71507928590176734735820650100, −4.35156331695660781084203995269, −4.32014485989271344474754956229, −4.21549869388628126670839612783, −4.00826063272024448316888000141, −3.94379316290337485469800444152, −3.89467604790867467242664013459, −3.52187322706236384613979336704, −3.33271733968894316057613519684, −3.27000335820518476310392028177, −3.10119958152249523559508703292, −2.62021621516129182697966224903, −2.52558297724396924021314255477, −2.40852046420157380343612050502, −2.22156312906962778224823103860, −2.19222594412830502245172157774, −1.92052261892594533998291353029, −1.70093403959821990369821741301, −1.59237771399940184716505464579, −1.07010449121753004343972994882, −1.03140266686704113044661630908, −0.68760119111977909916692903770, −0.62227424419970931131937199954, −0.47409032684847244214625537038, −0.37095865391295669053099305598, 0.37095865391295669053099305598, 0.47409032684847244214625537038, 0.62227424419970931131937199954, 0.68760119111977909916692903770, 1.03140266686704113044661630908, 1.07010449121753004343972994882, 1.59237771399940184716505464579, 1.70093403959821990369821741301, 1.92052261892594533998291353029, 2.19222594412830502245172157774, 2.22156312906962778224823103860, 2.40852046420157380343612050502, 2.52558297724396924021314255477, 2.62021621516129182697966224903, 3.10119958152249523559508703292, 3.27000335820518476310392028177, 3.33271733968894316057613519684, 3.52187322706236384613979336704, 3.89467604790867467242664013459, 3.94379316290337485469800444152, 4.00826063272024448316888000141, 4.21549869388628126670839612783, 4.32014485989271344474754956229, 4.35156331695660781084203995269, 4.71507928590176734735820650100

Graph of the ZZ-function along the critical line

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