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Properties

Label 2-3700-1.1-c1-0-10
Degree 22
Conductor 37003700
Sign 11
Analytic cond. 29.544629.5446
Root an. cond. 5.435495.43549
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s − 2·9-s − 3·11-s + 4·13-s − 4·19-s − 21-s + 5·27-s + 2·31-s + 3·33-s − 37-s − 4·39-s + 3·41-s − 2·43-s − 3·47-s − 6·49-s + 9·53-s + 4·57-s + 2·61-s − 2·63-s + 4·67-s + 15·71-s + 7·73-s − 3·77-s − 10·79-s + 81-s + 3·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.10·13-s − 0.917·19-s − 0.218·21-s + 0.962·27-s + 0.359·31-s + 0.522·33-s − 0.164·37-s − 0.640·39-s + 0.468·41-s − 0.304·43-s − 0.437·47-s − 6/7·49-s + 1.23·53-s + 0.529·57-s + 0.256·61-s − 0.251·63-s + 0.488·67-s + 1.78·71-s + 0.819·73-s − 0.341·77-s − 1.12·79-s + 1/9·81-s + 0.329·83-s + ⋯

Functional equation

Λ(s)=(3700s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3700s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 29.544629.5446
Root analytic conductor: 5.435495.43549
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3700, ( :1/2), 1)(2,\ 3700,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2234258791.223425879
L(12)L(\frac12) ≈\approx 1.2234258791.223425879
L(32)L(\frac{3}{2}) 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 1
5 1 1
37 1+T 1 + T
good3 1+T+pT2 1 + T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−15T+pT2 1 - 15 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−3T+pT2 1 - 3 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{2}
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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

−8.320566320344959555589170558707, −8.063386549062472212301129813398, −6.88610542134705476043905926803, −6.23303933687777995129974330697, −5.53161304223054046550311903611, −4.89610125391740010196132513501, −3.94769425088528710321117034658, −2.96736714350579454573712100915, −1.98523078966066915455111948391, −0.65226138130976440323790882395, 0.65226138130976440323790882395, 1.98523078966066915455111948391, 2.96736714350579454573712100915, 3.94769425088528710321117034658, 4.89610125391740010196132513501, 5.53161304223054046550311903611, 6.23303933687777995129974330697, 6.88610542134705476043905926803, 8.063386549062472212301129813398, 8.320566320344959555589170558707

Graph of the ZZ-function along the critical line