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Properties

Label 2-931-19.18-c0-0-1
Degree 22
Conductor 931931
Sign 11
Analytic cond. 0.4646290.464629
Root an. cond. 0.6816370.681637
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 5-s + 9-s − 11-s + 16-s − 2·17-s − 19-s + 20-s − 23-s + 36-s − 43-s − 44-s + 45-s + 47-s − 55-s + 61-s + 64-s − 2·68-s + 73-s − 76-s + 80-s + 81-s + 83-s − 2·85-s − 92-s − 95-s − 99-s + ⋯
L(s)  = 1  + 4-s + 5-s + 9-s − 11-s + 16-s − 2·17-s − 19-s + 20-s − 23-s + 36-s − 43-s − 44-s + 45-s + 47-s − 55-s + 61-s + 64-s − 2·68-s + 73-s − 76-s + 80-s + 81-s + 83-s − 2·85-s − 92-s − 95-s − 99-s + ⋯

Functional equation

Λ(s)=(931s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(931s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 931931    =    72⋅197^{2} \cdot 19
Sign: 11
Analytic conductor: 0.4646290.464629
Root analytic conductor: 0.6816370.681637
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ931(246,⋅)\chi_{931} (246, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 931, ( :0), 1)(2,\ 931,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3886110761.388611076
L(12)L(\frac12) ≈\approx 1.3886110761.388611076
L(1)L(1) 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)
bad7 1 1
19 1+T 1 + T
good2 (1−T)(1+T) ( 1 - T )( 1 + T )
3 (1−T)(1+T) ( 1 - T )( 1 + T )
5 1−T+T2 1 - T + T^{2}
11 1+T+T2 1 + T + T^{2}
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 (1+T)2 ( 1 + T )^{2}
23 1+T+T2 1 + T + T^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 1+T+T2 1 + T + T^{2}
47 1−T+T2 1 - T + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 1−T+T2 1 - T + T^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 1−T+T2 1 - T + T^{2}
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 1−T+T2 1 - T + T^{2}
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1−T)(1+T) ( 1 - T )( 1 + 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

−10.38739291117278531508092632646, −9.652843390388059780599493064835, −8.576381934259078022497488987091, −7.63384671755479415741608943405, −6.69331232280226476236494483754, −6.19908413004047436911737191485, −5.11240652066751133459345581876, −4.00502174264507225856117454247, −2.39726645069894493899591576473, −1.93189952504357432790521588434, 1.93189952504357432790521588434, 2.39726645069894493899591576473, 4.00502174264507225856117454247, 5.11240652066751133459345581876, 6.19908413004047436911737191485, 6.69331232280226476236494483754, 7.63384671755479415741608943405, 8.576381934259078022497488987091, 9.652843390388059780599493064835, 10.38739291117278531508092632646

Graph of the ZZ-function along the critical line