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Properties

Label 2-3700-148.147-c0-0-6
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 2.6913278772.691327877
L(12)L(\frac12) ≈\approx 2.6913278772.691327877
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

−8.438038724244311405770993811846, −7.88420378305325390075645747621, −6.99446324522723972533811963933, −6.42263979823660368993101691374, −5.75910621592356685767027135804, −4.59007463606938351211036862093, −4.36566813708242412163175731707, −3.33765426402307456970991998656, −2.36164885993655915262178801310, −1.42721421791403126229099306116, 1.42721421791403126229099306116, 2.36164885993655915262178801310, 3.33765426402307456970991998656, 4.36566813708242412163175731707, 4.59007463606938351211036862093, 5.75910621592356685767027135804, 6.42263979823660368993101691374, 6.99446324522723972533811963933, 7.88420378305325390075645747621, 8.438038724244311405770993811846

Graph of the ZZ-function along the critical line