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Properties

Label 2-37-37.10-c5-0-0
Degree $2$
Conductor $37$
Sign $0.126 + 0.992i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.15 + 8.92i)2-s + (13.8 + 23.9i)3-s + (−37.0 − 64.2i)4-s + (−33.8 − 58.6i)5-s − 284.·6-s + (−29.2 − 50.7i)7-s + 434.·8-s + (−260. + 450. i)9-s + 697.·10-s − 622.·11-s + (1.02e3 − 1.77e3i)12-s + (5.07 + 8.78i)13-s + 603.·14-s + (935. − 1.62e3i)15-s + (−1.04e3 + 1.81e3i)16-s + (−763. + 1.32e3i)17-s + ⋯
L(s)  = 1  + (−0.910 + 1.57i)2-s + (0.886 + 1.53i)3-s + (−1.15 − 2.00i)4-s + (−0.605 − 1.04i)5-s − 3.22·6-s + (−0.225 − 0.391i)7-s + 2.39·8-s + (−1.07 + 1.85i)9-s + 2.20·10-s − 1.55·11-s + (2.05 − 3.55i)12-s + (0.00832 + 0.0144i)13-s + 0.822·14-s + (1.07 − 1.86i)15-s + (−1.02 + 1.77i)16-s + (−0.640 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.126 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.126 + 0.992i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.126 + 0.992i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.277209 - 0.244178i\)
\(L(\frac12)\) \(\approx\) \(0.277209 - 0.244178i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-7.29e3 + 4.01e3i)T \)
good2 \( 1 + (5.15 - 8.92i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (-13.8 - 23.9i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (33.8 + 58.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (29.2 + 50.7i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 622.T + 1.61e5T^{2} \)
13 \( 1 + (-5.07 - 8.78i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (763. - 1.32e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-909. - 1.57e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 2.06e3T + 6.43e6T^{2} \)
29 \( 1 + 297.T + 2.05e7T^{2} \)
31 \( 1 + 2.74e3T + 2.86e7T^{2} \)
41 \( 1 + (-7.98e3 - 1.38e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 1.50e4T + 1.47e8T^{2} \)
47 \( 1 - 757.T + 2.29e8T^{2} \)
53 \( 1 + (-8.18e3 + 1.41e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.68e4 - 2.92e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.20e3 + 2.09e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.62e4 - 2.80e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-6.62e3 - 1.14e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 1.33e4T + 2.07e9T^{2} \)
79 \( 1 + (1.57e4 + 2.72e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (2.66e4 - 4.61e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (1.44e4 - 2.50e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.29e4T + 8.58e9T^{2} \)
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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

−16.23524051567473001263483827036, −15.49401348223623050993093719305, −14.63040251528142089801670011718, −13.27307635743183072541705187153, −10.50725807086272926754368799430, −9.650442067610951383083888049215, −8.416488676848816843293230725757, −7.922181153401913843305358678497, −5.45913512593111658340682216939, −4.22500414464519726815979822875, 0.24706223751558025308255884150, 2.37798992929540242012790547070, 3.04095184605867440315768339915, 7.20384932246986441351541372238, 8.041176094829465373501746453233, 9.325086448468625395232958946247, 10.91181086341245344964500262038, 11.90371683505514837349447870398, 12.97049044908421407413905268952, 13.82479759964969107991222408298

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