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Properties

Label 2-3700-148.147-c0-0-7
Degree 22
Conductor 37003700
Sign 11
Analytic cond. 1.846541.84654
Root an. cond. 1.358871.35887
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  + 2-s + 4-s + 8-s + 9-s + 16-s + 18-s + 19-s − 23-s − 2·31-s + 32-s + 36-s − 37-s + 38-s − 41-s − 43-s − 46-s + 49-s + 53-s + 59-s − 2·62-s + 64-s + 72-s + 73-s − 74-s + 76-s + 79-s + 81-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 9-s + 16-s + 18-s + 19-s − 23-s − 2·31-s + 32-s + 36-s − 37-s + 38-s − 41-s − 43-s − 46-s + 49-s + 53-s + 59-s − 2·62-s + 64-s + 72-s + 73-s − 74-s + 76-s + 79-s + 81-s + ⋯

Functional equation

Λ(s)=(3700s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(3700s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37003700    =    22⋅52⋅372^{2} \cdot 5^{2} \cdot 37
Sign: 11
Analytic conductor: 1.846541.84654
Root analytic conductor: 1.358871.35887
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ3700(3551,⋅)\chi_{3700} (3551, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3700, ( :0), 1)(2,\ 3700,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.7186503112.718650311
L(12)L(\frac12) ≈\approx 2.7186503112.718650311
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)
bad2 1−T 1 - T
5 1 1
37 1+T 1 + T
good3 (1−T)(1+T) ( 1 - T )( 1 + T )
7 (1−T)(1+T) ( 1 - T )( 1 + T )
11 (1−T)(1+T) ( 1 - T )( 1 + T )
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 1−T+T2 1 - T + T^{2}
23 1+T+T2 1 + T + T^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1+T)2 ( 1 + T )^{2}
41 1+T+T2 1 + T + T^{2}
43 1+T+T2 1 + T + T^{2}
47 (1−T)(1+T) ( 1 - T )( 1 + T )
53 1−T+T2 1 - T + T^{2}
59 1−T+T2 1 - T + T^{2}
61 (1−T)(1+T) ( 1 - T )( 1 + T )
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+T2 1 - T + T^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
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

−8.592937174691855517382500016546, −7.65094592422006091657213230332, −7.14249785478647335730961813380, −6.49165200467896357366146831755, −5.46800243957701479669054008505, −5.05397676496170003061080747826, −3.91413153262908031228894475825, −3.57014442408487516126598953603, −2.29006192310136176306588255000, −1.45346304321950118282171325050, 1.45346304321950118282171325050, 2.29006192310136176306588255000, 3.57014442408487516126598953603, 3.91413153262908031228894475825, 5.05397676496170003061080747826, 5.46800243957701479669054008505, 6.49165200467896357366146831755, 7.14249785478647335730961813380, 7.65094592422006091657213230332, 8.592937174691855517382500016546

Graph of the ZZ-function along the critical line