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Properties

Label 2-59-59.58-c12-0-22
Degree 22
Conductor 5959
Sign 11
Analytic cond. 53.925653.9256
Root an. cond. 7.343407.34340
Motivic weight 1212
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.43e3·3-s + 4.09e3·4-s + 5.23e3·5-s − 2.11e5·7-s + 1.52e6·9-s − 5.86e6·12-s − 7.49e6·15-s + 1.67e7·16-s − 1.16e7·17-s − 9.38e7·19-s + 2.14e7·20-s + 3.02e8·21-s − 2.16e8·25-s − 1.41e9·27-s − 8.65e8·28-s − 6.37e8·29-s − 1.10e9·35-s + 6.23e9·36-s + 9.48e9·41-s + 7.96e9·45-s − 2.40e10·48-s + 3.07e10·49-s + 1.67e10·51-s − 7.91e9·53-s + 1.34e11·57-s + 4.21e10·59-s − 3.07e10·60-s + ⋯
L(s)  = 1  − 1.96·3-s + 4-s + 0.334·5-s − 1.79·7-s + 2.86·9-s − 1.96·12-s − 0.658·15-s + 16-s − 0.483·17-s − 1.99·19-s + 0.334·20-s + 3.52·21-s − 0.887·25-s − 3.66·27-s − 1.79·28-s − 1.07·29-s − 0.601·35-s + 2.86·36-s + 1.99·41-s + 0.958·45-s − 1.96·48-s + 2.22·49-s + 0.950·51-s − 0.357·53-s + 3.92·57-s + 59-s − 0.658·60-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=( Λ(13−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+6) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: 11
Analytic conductor: 53.925653.9256
Root analytic conductor: 7.343407.34340
Motivic weight: 1212
Rational: yes
Arithmetic: yes
Character: χ59(58,⋅)\chi_{59} (58, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 59, ( :6), 1)(2,\ 59,\ (\ :6),\ 1)

Particular Values

L(132)L(\frac{13}{2}) ≈\approx 0.67421890390.6742189039
L(12)L(\frac12) ≈\approx 0.67421890390.6742189039
L(7)L(7) 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)
bad59 1−p6T 1 - p^{6} T
good2 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
3 1+1433T+p12T2 1 + 1433 T + p^{12} T^{2}
5 1−5231T+p12T2 1 - 5231 T + p^{12} T^{2}
7 1+211273T+p12T2 1 + 211273 T + p^{12} T^{2}
11 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
13 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
17 1+11672638T+p12T2 1 + 11672638 T + p^{12} T^{2}
19 1+93895513T+p12T2 1 + 93895513 T + p^{12} T^{2}
23 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
29 1+637537633T+p12T2 1 + 637537633 T + p^{12} T^{2}
31 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
37 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
41 1−9483946607T+p12T2 1 - 9483946607 T + p^{12} T^{2}
43 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
47 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
53 1+7914541633T+p12T2 1 + 7914541633 T + p^{12} T^{2}
61 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
67 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
71 1−210865062242T+p12T2 1 - 210865062242 T + p^{12} T^{2}
73 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
79 1+402738855433T+p12T2 1 + 402738855433 T + p^{12} T^{2}
83 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
89 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
97 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.60105779640211405362670502795, −11.34643054560229609511947925050, −10.53721206325074530033506681007, −9.661277348731806319473336445094, −7.14134374636648922442822990565, −6.30430258635853045707634997246, −5.83678605388361622782114589238, −4.01044719546719501778210240220, −2.09729684936529905219395962876, −0.47526207818152328414793319320, 0.47526207818152328414793319320, 2.09729684936529905219395962876, 4.01044719546719501778210240220, 5.83678605388361622782114589238, 6.30430258635853045707634997246, 7.14134374636648922442822990565, 9.661277348731806319473336445094, 10.53721206325074530033506681007, 11.34643054560229609511947925050, 12.60105779640211405362670502795

Graph of the ZZ-function along the critical line