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Properties

Label 12-930e6-1.1-c1e6-0-2
Degree 1212
Conductor 6.470×10176.470\times 10^{17}
Sign 11
Analytic cond. 167710.167710.
Root an. cond. 2.725082.72508
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  − 3·4-s − 4·5-s − 3·9-s + 16·11-s + 6·16-s + 4·19-s + 12·20-s + 4·25-s + 4·29-s − 6·31-s + 9·36-s + 12·41-s − 48·44-s + 12·45-s + 30·49-s − 64·55-s + 4·59-s + 28·61-s − 10·64-s − 16·71-s − 12·76-s + 12·79-s − 24·80-s + 6·81-s + 4·89-s − 16·95-s − 48·99-s + ⋯
L(s)  = 1  − 3/2·4-s − 1.78·5-s − 9-s + 4.82·11-s + 3/2·16-s + 0.917·19-s + 2.68·20-s + 4/5·25-s + 0.742·29-s − 1.07·31-s + 3/2·36-s + 1.87·41-s − 7.23·44-s + 1.78·45-s + 30/7·49-s − 8.62·55-s + 0.520·59-s + 3.58·61-s − 5/4·64-s − 1.89·71-s − 1.37·76-s + 1.35·79-s − 2.68·80-s + 2/3·81-s + 0.423·89-s − 1.64·95-s − 4.82·99-s + ⋯

Functional equation

Λ(s)=((26⋅36⋅56⋅316)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((26⋅36⋅56⋅316)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 26⋅36⋅56⋅3162^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}
Sign: 11
Analytic conductor: 167710.167710.
Root analytic conductor: 2.725082.72508
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 26⋅36⋅56⋅316, ( :[1/2]6), 1)(12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.9921618613.992161861
L(12)L(\frac12) ≈\approx 3.9921618613.992161861
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+T2)3 ( 1 + T^{2} )^{3}
3 (1+T2)3 ( 1 + T^{2} )^{3}
5 1+4T+12T2+36T3+12pT4+4p2T5+p3T6 1 + 4 T + 12 T^{2} + 36 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
31 (1+T)6 ( 1 + T )^{6}
good7 (1−10T2+p2T4)3 ( 1 - 10 T^{2} + p^{2} T^{4} )^{3}
11 (1−8T+46T2−168T3+46pT4−8p2T5+p3T6)2 ( 1 - 8 T + 46 T^{2} - 168 T^{3} + 46 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
13 1−28T2+280T4−2014T6+280p2T8−28p4T10+p6T12 1 - 28 T^{2} + 280 T^{4} - 2014 T^{6} + 280 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12}
17 1−28T2+512T4−11034T6+512p2T8−28p4T10+p6T12 1 - 28 T^{2} + 512 T^{4} - 11034 T^{6} + 512 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12}
19 (1−2T+22T2−80T3+22pT4−2p2T5+p3T6)2 ( 1 - 2 T + 22 T^{2} - 80 T^{3} + 22 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
23 1−70T2+2639T4−71412T6+2639p2T8−70p4T10+p6T12 1 - 70 T^{2} + 2639 T^{4} - 71412 T^{6} + 2639 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12}
29 (1−2T+55T2−132T3+55pT4−2p2T5+p3T6)2 ( 1 - 2 T + 55 T^{2} - 132 T^{3} + 55 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 1−58T2+1895T4−66348T6+1895p2T8−58p4T10+p6T12 1 - 58 T^{2} + 1895 T^{4} - 66348 T^{6} + 1895 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12}
41 (1−2T+pT2)6 ( 1 - 2 T + p T^{2} )^{6}
43 1−106T2+7367T4−400332T6+7367p2T8−106p4T10+p6T12 1 - 106 T^{2} + 7367 T^{4} - 400332 T^{6} + 7367 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12}
47 1−208T2+20432T4−1209294T6+20432p2T8−208p4T10+p6T12 1 - 208 T^{2} + 20432 T^{4} - 1209294 T^{6} + 20432 p^{2} T^{8} - 208 p^{4} T^{10} + p^{6} T^{12}
53 1−130T2+4055T4+2820T6+4055p2T8−130p4T10+p6T12 1 - 130 T^{2} + 4055 T^{4} + 2820 T^{6} + 4055 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12}
59 (1−2T+101T2+44T3+101pT4−2p2T5+p3T6)2 ( 1 - 2 T + 101 T^{2} + 44 T^{3} + 101 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1−14T+92T2−336T3+92pT4−14p2T5+p3T6)2 ( 1 - 14 T + 92 T^{2} - 336 T^{3} + 92 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 1−116T2+11776T4−712958T6+11776p2T8−116p4T10+p6T12 1 - 116 T^{2} + 11776 T^{4} - 712958 T^{6} + 11776 p^{2} T^{8} - 116 p^{4} T^{10} + p^{6} T^{12}
71 (1+8T+134T2+1156T3+134pT4+8p2T5+p3T6)2 ( 1 + 8 T + 134 T^{2} + 1156 T^{3} + 134 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−170T2+20671T4−1830476T6+20671p2T8−170p4T10+p6T12 1 - 170 T^{2} + 20671 T^{4} - 1830476 T^{6} + 20671 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12}
79 (1−6T+230T2−920T3+230pT4−6p2T5+p3T6)2 ( 1 - 6 T + 230 T^{2} - 920 T^{3} + 230 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 1−128T2+13008T4−1479782T6+13008p2T8−128p4T10+p6T12 1 - 128 T^{2} + 13008 T^{4} - 1479782 T^{6} + 13008 p^{2} T^{8} - 128 p^{4} T^{10} + p^{6} T^{12}
89 (1−2T+123T2−36T3+123pT4−2p2T5+p3T6)2 ( 1 - 2 T + 123 T^{2} - 36 T^{3} + 123 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
97 1−432T2+89120T4−10929554T6+89120p2T8−432p4T10+p6T12 1 - 432 T^{2} + 89120 T^{4} - 10929554 T^{6} + 89120 p^{2} T^{8} - 432 p^{4} T^{10} + 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

−5.28579331807154264471005186648, −5.17898751134358155285796112113, −5.05774721293566686167443644511, −4.82320216324444104996066113937, −4.50416628739632516664040818881, −4.39630658267190707424944343763, −4.15923503549313541264775914404, −4.02776596307716053345255355874, −4.01845292749549910138937693410, −3.77988037861326749413454058666, −3.76122966367400274574582673828, −3.72840688322061401172338255051, −3.66575857188705674443097216072, −2.99862139772311652847097024544, −2.94188179586906768208890435097, −2.82763970041976481777680247682, −2.73154872617466493573894720637, −2.09185019993051032073452075394, −2.04695849668419524796122432955, −1.63117750483127575910222366453, −1.57738892623066139293423674829, −1.01461484730051513890731435765, −0.78776609613215060268633625533, −0.75028415678710255794022844439, −0.53985296122424655334859698180, 0.53985296122424655334859698180, 0.75028415678710255794022844439, 0.78776609613215060268633625533, 1.01461484730051513890731435765, 1.57738892623066139293423674829, 1.63117750483127575910222366453, 2.04695849668419524796122432955, 2.09185019993051032073452075394, 2.73154872617466493573894720637, 2.82763970041976481777680247682, 2.94188179586906768208890435097, 2.99862139772311652847097024544, 3.66575857188705674443097216072, 3.72840688322061401172338255051, 3.76122966367400274574582673828, 3.77988037861326749413454058666, 4.01845292749549910138937693410, 4.02776596307716053345255355874, 4.15923503549313541264775914404, 4.39630658267190707424944343763, 4.50416628739632516664040818881, 4.82320216324444104996066113937, 5.05774721293566686167443644511, 5.17898751134358155285796112113, 5.28579331807154264471005186648

Graph of the ZZ-function along the critical line

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