Properties

Label 2-37-1.1-c9-0-25
Degree $2$
Conductor $37$
Sign $-1$
Analytic cond. $19.0563$
Root an. cond. $4.36535$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.7·2-s + 116.·3-s − 79.6·4-s − 29.0·5-s + 2.42e3·6-s − 1.09e4·7-s − 1.23e4·8-s − 6.11e3·9-s − 603.·10-s + 1.40e4·11-s − 9.28e3·12-s − 3.90e4·13-s − 2.26e5·14-s − 3.37e3·15-s − 2.14e5·16-s − 3.16e5·17-s − 1.27e5·18-s + 8.56e5·19-s + 2.31e3·20-s − 1.27e6·21-s + 2.92e5·22-s + 2.51e6·23-s − 1.43e6·24-s − 1.95e6·25-s − 8.12e5·26-s − 3.00e6·27-s + 8.69e5·28-s + ⋯
L(s)  = 1  + 0.918·2-s + 0.830·3-s − 0.155·4-s − 0.0207·5-s + 0.762·6-s − 1.71·7-s − 1.06·8-s − 0.310·9-s − 0.0190·10-s + 0.290·11-s − 0.129·12-s − 0.379·13-s − 1.57·14-s − 0.0172·15-s − 0.820·16-s − 0.919·17-s − 0.285·18-s + 1.50·19-s + 0.00323·20-s − 1.42·21-s + 0.266·22-s + 1.87·23-s − 0.881·24-s − 0.999·25-s − 0.348·26-s − 1.08·27-s + 0.267·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-1$
Analytic conductor: \(19.0563\)
Root analytic conductor: \(4.36535\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 37,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + 1.87e6T \)
good2 \( 1 - 20.7T + 512T^{2} \)
3 \( 1 - 116.T + 1.96e4T^{2} \)
5 \( 1 + 29.0T + 1.95e6T^{2} \)
7 \( 1 + 1.09e4T + 4.03e7T^{2} \)
11 \( 1 - 1.40e4T + 2.35e9T^{2} \)
13 \( 1 + 3.90e4T + 1.06e10T^{2} \)
17 \( 1 + 3.16e5T + 1.18e11T^{2} \)
19 \( 1 - 8.56e5T + 3.22e11T^{2} \)
23 \( 1 - 2.51e6T + 1.80e12T^{2} \)
29 \( 1 + 3.16e6T + 1.45e13T^{2} \)
31 \( 1 + 3.02e6T + 2.64e13T^{2} \)
41 \( 1 + 7.28e6T + 3.27e14T^{2} \)
43 \( 1 + 3.46e7T + 5.02e14T^{2} \)
47 \( 1 - 1.46e7T + 1.11e15T^{2} \)
53 \( 1 - 5.03e7T + 3.29e15T^{2} \)
59 \( 1 - 1.17e8T + 8.66e15T^{2} \)
61 \( 1 + 1.39e8T + 1.16e16T^{2} \)
67 \( 1 - 1.76e8T + 2.72e16T^{2} \)
71 \( 1 + 6.58e7T + 4.58e16T^{2} \)
73 \( 1 + 1.77e8T + 5.88e16T^{2} \)
79 \( 1 + 2.52e8T + 1.19e17T^{2} \)
83 \( 1 + 7.87e8T + 1.86e17T^{2} \)
89 \( 1 - 8.77e8T + 3.50e17T^{2} \)
97 \( 1 + 1.35e9T + 7.60e17T^{2} \)
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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

−13.55111685642879480488643057593, −13.08647198152043912157961450771, −11.71804556166452681114776434256, −9.622506797699085921880460581541, −8.946456390729419901102896867786, −6.96693469374191736841612054891, −5.51138408430608438116007424916, −3.67176701087675617548585023748, −2.85414154134567379323323563665, 0, 2.85414154134567379323323563665, 3.67176701087675617548585023748, 5.51138408430608438116007424916, 6.96693469374191736841612054891, 8.946456390729419901102896867786, 9.622506797699085921880460581541, 11.71804556166452681114776434256, 13.08647198152043912157961450771, 13.55111685642879480488643057593

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