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Properties

Label 2-37-37.26-c5-0-12
Degree $2$
Conductor $37$
Sign $-0.999 + 0.0260i$
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  + (−3.72 − 6.44i)2-s + (5.98 − 10.3i)3-s + (−11.7 + 20.3i)4-s + (37.3 − 64.7i)5-s − 89.0·6-s + (68.5 − 118. i)7-s − 63.6·8-s + (49.9 + 86.4i)9-s − 556.·10-s + 109.·11-s + (140. + 242. i)12-s + (−500. + 867. i)13-s − 1.02e3·14-s + (−447. − 774. i)15-s + (612. + 1.06e3i)16-s + (−451. − 782. i)17-s + ⋯
L(s)  = 1  + (−0.658 − 1.13i)2-s + (0.383 − 0.664i)3-s + (−0.366 + 0.634i)4-s + (0.668 − 1.15i)5-s − 1.01·6-s + (0.528 − 0.916i)7-s − 0.351·8-s + (0.205 + 0.355i)9-s − 1.76·10-s + 0.273·11-s + (0.281 + 0.487i)12-s + (−0.821 + 1.42i)13-s − 1.39·14-s + (−0.513 − 0.888i)15-s + (0.597 + 1.03i)16-s + (−0.378 − 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0260i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.999 + 0.0260i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.999 + 0.0260i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0179852 - 1.37950i\)
\(L(\frac12)\) \(\approx\) \(0.0179852 - 1.37950i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.85e3 + 7.82e3i)T \)
good2 \( 1 + (3.72 + 6.44i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-5.98 + 10.3i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-37.3 + 64.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-68.5 + 118. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 109.T + 1.61e5T^{2} \)
13 \( 1 + (500. - 867. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (451. + 782. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-425. + 736. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 - 1.60e3T + 6.43e6T^{2} \)
29 \( 1 - 6.17e3T + 2.05e7T^{2} \)
31 \( 1 + 8.27e3T + 2.86e7T^{2} \)
41 \( 1 + (3.99e3 - 6.91e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + 3.86e3T + 1.47e8T^{2} \)
47 \( 1 - 1.72e4T + 2.29e8T^{2} \)
53 \( 1 + (1.42e4 + 2.46e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.35e4 - 2.35e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.12e4 - 1.94e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.72e4 + 4.71e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (299. - 518. i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 - 7.98e3T + 2.07e9T^{2} \)
79 \( 1 + (6.65e3 - 1.15e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-560. - 970. i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (4.07e4 + 7.06e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.06e5T + 8.58e9T^{2} \)
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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

−14.27662397880905708106751698018, −13.33766775574281797841502473948, −12.27246348887241484948823413285, −11.05579750741703344772710605306, −9.634972292689241045105417260130, −8.762036478227360597808763856005, −7.13608889258309264544040933981, −4.69959170932300880826192062155, −2.09749675946845700211834705431, −1.02778208004052291149870132202, 2.92330803740810421236419557713, 5.55332704361319237741501180252, 6.83579299996658927794778789256, 8.289536184135038896125037546884, 9.481411541722360526689828042016, 10.50919544039693924265933419828, 12.34280046563001519641311860098, 14.41231074241778914930537400330, 15.00395423035984128635177799106, 15.60805001228859061179848210221

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