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Properties

Label 2-59-59.58-c6-0-16
Degree 22
Conductor 5959
Sign 11
Analytic cond. 13.573113.5731
Root an. cond. 3.684183.68418
Motivic weight 66
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s + 64·4-s + 191·5-s + 155·7-s − 704·9-s − 320·12-s − 955·15-s + 4.09e3·16-s + 6.05e3·17-s + 443·19-s + 1.22e4·20-s − 775·21-s + 2.08e4·25-s + 7.16e3·27-s + 9.92e3·28-s − 2.34e4·29-s + 2.96e4·35-s − 4.50e4·36-s + 1.37e5·41-s − 1.34e5·45-s − 2.04e4·48-s − 9.36e4·49-s − 3.02e4·51-s − 1.90e5·53-s − 2.21e3·57-s − 2.05e5·59-s − 6.11e4·60-s + ⋯
L(s)  = 1  − 0.185·3-s + 4-s + 1.52·5-s + 0.451·7-s − 0.965·9-s − 0.185·12-s − 0.282·15-s + 16-s + 1.23·17-s + 0.0645·19-s + 1.52·20-s − 0.0836·21-s + 1.33·25-s + 0.364·27-s + 0.451·28-s − 0.963·29-s + 0.690·35-s − 0.965·36-s + 1.99·41-s − 1.47·45-s − 0.185·48-s − 0.795·49-s − 0.228·51-s − 1.28·53-s − 0.0119·57-s − 59-s − 0.282·60-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: 11
Analytic conductor: 13.573113.5731
Root analytic conductor: 3.684183.68418
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: χ59(58,⋅)\chi_{59} (58, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 59, ( :3), 1)(2,\ 59,\ (\ :3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 2.7370335522.737033552
L(12)L(\frac12) ≈\approx 2.7370335522.737033552
L(4)L(4) 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+p3T 1 + p^{3} T
good2 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
3 1+5T+p6T2 1 + 5 T + p^{6} T^{2}
5 1−191T+p6T2 1 - 191 T + p^{6} T^{2}
7 1−155T+p6T2 1 - 155 T + p^{6} T^{2}
11 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
13 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
17 1−6050T+p6T2 1 - 6050 T + p^{6} T^{2}
19 1−443T+p6T2 1 - 443 T + p^{6} T^{2}
23 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
29 1+23497T+p6T2 1 + 23497 T + p^{6} T^{2}
31 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
37 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
41 1−137783T+p6T2 1 - 137783 T + p^{6} T^{2}
43 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
47 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
53 1+190825T+p6T2 1 + 190825 T + p^{6} T^{2}
61 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
67 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
71 1+683422T+p6T2 1 + 683422 T + p^{6} T^{2}
73 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
79 1+288853T+p6T2 1 + 288853 T + p^{6} T^{2}
83 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
89 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
97 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} 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

−14.10584551369521964542591014881, −12.70824428199690825193982961526, −11.48717184455605153560599841218, −10.52034171822121839564032932065, −9.355985583408853208116101962119, −7.75698367492750124051403542193, −6.18499513177944521331361087885, −5.46802252947789690290386787815, −2.82786542540259953009667312199, −1.51537091767807603673283425594, 1.51537091767807603673283425594, 2.82786542540259953009667312199, 5.46802252947789690290386787815, 6.18499513177944521331361087885, 7.75698367492750124051403542193, 9.355985583408853208116101962119, 10.52034171822121839564032932065, 11.48717184455605153560599841218, 12.70824428199690825193982961526, 14.10584551369521964542591014881

Graph of the ZZ-function along the critical line