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Properties

Label 2-37030-1.1-c1-0-7
Degree $2$
Conductor $37030$
Sign $1$
Analytic cond. $295.686$
Root an. cond. $17.1955$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 3·9-s − 10-s + 4·13-s − 14-s + 16-s − 4·17-s + 3·18-s + 2·19-s + 20-s + 25-s − 4·26-s + 28-s + 10·29-s − 6·31-s − 32-s + 4·34-s + 35-s − 3·36-s + 6·37-s − 2·38-s − 40-s − 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.458·19-s + 0.223·20-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.85·29-s − 1.07·31-s − 0.176·32-s + 0.685·34-s + 0.169·35-s − 1/2·36-s + 0.986·37-s − 0.324·38-s − 0.158·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37030\)    =    \(2 \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(295.686\)
Root analytic conductor: \(17.1955\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 37030,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.794568066\)
\(L(\frac12)\) \(\approx\) \(1.794568066\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.77938834106085, −14.41561421345212, −13.92609005025255, −13.22650255388298, −12.97950472695069, −11.98128330293627, −11.64544193929041, −11.07362351945189, −10.75819529803028, −10.08278074980029, −9.499826889834329, −8.899487196610573, −8.449910193714145, −8.182075408909109, −7.323627061436514, −6.661798869860078, −6.234665443400454, −5.607915836519983, −5.060427044834532, −4.228125498667744, −3.470038808270815, −2.726021760672783, −2.182793881217394, −1.318105677110777, −0.5885105389949281, 0.5885105389949281, 1.318105677110777, 2.182793881217394, 2.726021760672783, 3.470038808270815, 4.228125498667744, 5.060427044834532, 5.607915836519983, 6.234665443400454, 6.661798869860078, 7.323627061436514, 8.182075408909109, 8.449910193714145, 8.899487196610573, 9.499826889834329, 10.08278074980029, 10.75819529803028, 11.07362351945189, 11.64544193929041, 11.98128330293627, 12.97950472695069, 13.22650255388298, 13.92609005025255, 14.41561421345212, 14.77938834106085

Graph of the $Z$-function along the critical line