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Properties

Label 2-37-37.31-c2-0-5
Degree $2$
Conductor $37$
Sign $-0.997 - 0.0655i$
Analytic cond. $1.00817$
Root an. cond. $1.00408$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.74 − 2.74i)2-s − 4.56i·3-s + 11.0i·4-s + (0.592 − 0.592i)5-s + (−12.5 + 12.5i)6-s − 2.28·7-s + (19.4 − 19.4i)8-s − 11.8·9-s − 3.25·10-s − 8.29i·11-s + 50.6·12-s + (3.30 − 3.30i)13-s + (6.28 + 6.28i)14-s + (−2.70 − 2.70i)15-s − 62.5·16-s + (−5.43 + 5.43i)17-s + ⋯
L(s)  = 1  + (−1.37 − 1.37i)2-s − 1.52i·3-s + 2.77i·4-s + (0.118 − 0.118i)5-s + (−2.08 + 2.08i)6-s − 0.326·7-s + (2.43 − 2.43i)8-s − 1.31·9-s − 0.325·10-s − 0.754i·11-s + 4.21·12-s + (0.253 − 0.253i)13-s + (0.448 + 0.448i)14-s + (−0.180 − 0.180i)15-s − 3.91·16-s + (−0.319 + 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.997 - 0.0655i$
Analytic conductor: \(1.00817\)
Root analytic conductor: \(1.00408\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1),\ -0.997 - 0.0655i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0170302 + 0.518964i\)
\(L(\frac12)\) \(\approx\) \(0.0170302 + 0.518964i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-29.7 + 22.0i)T \)
good2 \( 1 + (2.74 + 2.74i)T + 4iT^{2} \)
3 \( 1 + 4.56iT - 9T^{2} \)
5 \( 1 + (-0.592 + 0.592i)T - 25iT^{2} \)
7 \( 1 + 2.28T + 49T^{2} \)
11 \( 1 + 8.29iT - 121T^{2} \)
13 \( 1 + (-3.30 + 3.30i)T - 169iT^{2} \)
17 \( 1 + (5.43 - 5.43i)T - 289iT^{2} \)
19 \( 1 + (-15.5 + 15.5i)T - 361iT^{2} \)
23 \( 1 + (-15.0 + 15.0i)T - 529iT^{2} \)
29 \( 1 + (-4.60 - 4.60i)T + 841iT^{2} \)
31 \( 1 + (-36.3 - 36.3i)T + 961iT^{2} \)
41 \( 1 - 4.50iT - 1.68e3T^{2} \)
43 \( 1 + (-24.2 + 24.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 29.6T + 2.20e3T^{2} \)
53 \( 1 + 23.5T + 2.80e3T^{2} \)
59 \( 1 + (56.8 - 56.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (-43.1 - 43.1i)T + 3.72e3iT^{2} \)
67 \( 1 - 109. iT - 4.48e3T^{2} \)
71 \( 1 - 46.2T + 5.04e3T^{2} \)
73 \( 1 + 19.8iT - 5.32e3T^{2} \)
79 \( 1 + (-66.5 + 66.5i)T - 6.24e3iT^{2} \)
83 \( 1 - 46.6T + 6.88e3T^{2} \)
89 \( 1 + (-42.1 - 42.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (86.1 - 86.1i)T - 9.40e3iT^{2} \)
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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

−16.22242755151526972721463101403, −13.53599606018150816417766394592, −12.85974314613514510401724463803, −11.81119629678272880768754782578, −10.77787511974169189739376692430, −9.148222699186938379316926578123, −8.107685180386171087929999421531, −6.86045877200766252916478029808, −2.88123668655514617087825592058, −1.02756784431474061476768627988, 4.82004970103758044785080306114, 6.33634479759904947366924745360, 7.991767175047825905764401703873, 9.529042441731512880022834757946, 9.823234303789044219296816513136, 11.13648729891178664491141394810, 14.03410746169748264859606540115, 15.10238314716960976643068629165, 15.79282515815384416506273207649, 16.56678115004631413454146256379

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