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Properties

Label 2-37-37.6-c6-0-11
Degree $2$
Conductor $37$
Sign $0.979 + 0.203i$
Analytic cond. $8.51200$
Root an. cond. $2.91753$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.60 − 8.60i)2-s + 22.2i·3-s − 84.2i·4-s + (144. + 144. i)5-s + (191. + 191. i)6-s − 94.2·7-s + (−174. − 174. i)8-s + 233.·9-s + 2.49e3·10-s − 546. i·11-s + 1.87e3·12-s + (60.3 + 60.3i)13-s + (−811. + 811. i)14-s + (−3.21e3 + 3.21e3i)15-s + 2.38e3·16-s + (−3.06e3 − 3.06e3i)17-s + ⋯
L(s)  = 1  + (1.07 − 1.07i)2-s + 0.824i·3-s − 1.31i·4-s + (1.15 + 1.15i)5-s + (0.886 + 0.886i)6-s − 0.274·7-s + (−0.340 − 0.340i)8-s + 0.320·9-s + 2.49·10-s − 0.410i·11-s + 1.08·12-s + (0.0274 + 0.0274i)13-s + (−0.295 + 0.295i)14-s + (−0.953 + 0.953i)15-s + 0.583·16-s + (−0.623 − 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.979 + 0.203i$
Analytic conductor: \(8.51200\)
Root analytic conductor: \(2.91753\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3),\ 0.979 + 0.203i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.36287 - 0.346091i\)
\(L(\frac12)\) \(\approx\) \(3.36287 - 0.346091i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (3.11e4 + 3.99e4i)T \)
good2 \( 1 + (-8.60 + 8.60i)T - 64iT^{2} \)
3 \( 1 - 22.2iT - 729T^{2} \)
5 \( 1 + (-144. - 144. i)T + 1.56e4iT^{2} \)
7 \( 1 + 94.2T + 1.17e5T^{2} \)
11 \( 1 + 546. iT - 1.77e6T^{2} \)
13 \( 1 + (-60.3 - 60.3i)T + 4.82e6iT^{2} \)
17 \( 1 + (3.06e3 + 3.06e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (4.52e3 + 4.52e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (161. + 161. i)T + 1.48e8iT^{2} \)
29 \( 1 + (-1.55e4 + 1.55e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (2.77e4 - 2.77e4i)T - 8.87e8iT^{2} \)
41 \( 1 + 7.79e4iT - 4.75e9T^{2} \)
43 \( 1 + (-4.01e4 - 4.01e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 1.59e5T + 1.07e10T^{2} \)
53 \( 1 - 1.80e5T + 2.21e10T^{2} \)
59 \( 1 + (2.45e5 + 2.45e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.76e5 + 1.76e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 1.21e5iT - 9.04e10T^{2} \)
71 \( 1 - 6.98e5T + 1.28e11T^{2} \)
73 \( 1 - 2.43e5iT - 1.51e11T^{2} \)
79 \( 1 + (2.85e5 + 2.85e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 6.12e4T + 3.26e11T^{2} \)
89 \( 1 + (3.39e5 - 3.39e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-6.15e5 - 6.15e5i)T + 8.32e11iT^{2} \)
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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

−14.63081072426267761558846785950, −13.81354917160800375792626450457, −12.83162138560030875932682341980, −11.12689981133974718868292470540, −10.49041504891862787105736210939, −9.448637796244906385182428639222, −6.61246993877782563005219021298, −5.08424429079125908543393698658, −3.51471690717063839777672564618, −2.22437244745217586787800159991, 1.61531376732508941426062429923, 4.47023202911911658068850313141, 5.81731188428675555949685112312, 6.79764211469954903527607170278, 8.326879319802278553576901095199, 9.941602472766243905709345633708, 12.47882852616842730316642458287, 12.98047020879393293205008424505, 13.71504093680406582366664706254, 14.99531149574205083681424670301

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