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Properties

Label 2-37-37.4-c1-0-0
Degree $2$
Conductor $37$
Sign $0.665 - 0.746i$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.59 − 0.456i)2-s + (0.354 + 2.00i)3-s + (4.62 + 1.68i)4-s + (0.640 + 0.763i)5-s − 5.36i·6-s + (−1.58 + 1.33i)7-s + (−6.65 − 3.84i)8-s + (−1.09 + 0.398i)9-s + (−1.31 − 2.27i)10-s + (0.884 − 1.53i)11-s + (−1.74 + 9.88i)12-s + (1.83 − 5.05i)13-s + (4.71 − 2.72i)14-s + (−1.30 + 1.55i)15-s + (7.93 + 6.66i)16-s + (0.448 + 1.23i)17-s + ⋯
L(s)  = 1  + (−1.83 − 0.322i)2-s + (0.204 + 1.16i)3-s + (2.31 + 0.841i)4-s + (0.286 + 0.341i)5-s − 2.19i·6-s + (−0.599 + 0.502i)7-s + (−2.35 − 1.35i)8-s + (−0.364 + 0.132i)9-s + (−0.414 − 0.718i)10-s + (0.266 − 0.462i)11-s + (−0.503 + 2.85i)12-s + (0.509 − 1.40i)13-s + (1.26 − 0.727i)14-s + (−0.337 + 0.402i)15-s + (1.98 + 1.66i)16-s + (0.108 + 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365079 + 0.163696i\)
\(L(\frac12)\) \(\approx\) \(0.365079 + 0.163696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (6.03 + 0.746i)T \)
good2 \( 1 + (2.59 + 0.456i)T + (1.87 + 0.684i)T^{2} \)
3 \( 1 + (-0.354 - 2.00i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-0.640 - 0.763i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.58 - 1.33i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.884 + 1.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.83 + 5.05i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.448 - 1.23i)T + (-13.0 + 10.9i)T^{2} \)
19 \( 1 + (4.12 - 0.726i)T + (17.8 - 6.49i)T^{2} \)
23 \( 1 + (-3.05 + 1.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.525 + 0.303i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.55iT - 31T^{2} \)
41 \( 1 + (-2.15 - 0.782i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 8.37iT - 43T^{2} \)
47 \( 1 + (1.79 + 3.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.47 + 7.11i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (6.76 - 8.06i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.41 - 3.89i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-3.77 + 3.16i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.05 + 6.00i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + (8.59 + 10.2i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.39 + 0.507i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (4.42 - 5.27i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (6.55 - 3.78i)T + (48.5 - 84.0i)T^{2} \)
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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

−16.65874865632341910078903272198, −15.80600018004111820414176484078, −14.95297266340316335349170835141, −12.62203732497551915640624384251, −10.90858805223402129549855456320, −10.27559179560686029677880561259, −9.252640275806639759625928643726, −8.254119974705986084459834408277, −6.31389571233279189448426125956, −3.10358972461483080333449386348, 1.66714993466177284440181113813, 6.57677545816096445250287640972, 7.21571006289351911073823198259, 8.677065178148559390991995525512, 9.646115689926499324566757883277, 11.11012468205327772158796646407, 12.55519325717131973598136445411, 13.93077739884186839297363975542, 15.62783214487590836095813740686, 16.80916760846374611976723166161

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