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Properties

Label 2-37-37.36-c7-0-12
Degree $2$
Conductor $37$
Sign $0.454 - 0.890i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.0i·2-s + 52.6·3-s − 128.·4-s − 434. i·5-s + 842. i·6-s + 1.47e3·7-s − 0.510i·8-s + 585.·9-s + 6.95e3·10-s + 4.55e3·11-s − 6.74e3·12-s − 4.82e3i·13-s + 2.36e4i·14-s − 2.29e4i·15-s − 1.63e4·16-s + 3.56e3i·17-s + ⋯
L(s)  = 1  + 1.41i·2-s + 1.12·3-s − 1.00·4-s − 1.55i·5-s + 1.59i·6-s + 1.62·7-s − 0.000352i·8-s + 0.267·9-s + 2.20·10-s + 1.03·11-s − 1.12·12-s − 0.609i·13-s + 2.30i·14-s − 1.75i·15-s − 0.999·16-s + 0.176i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.47232 + 1.51493i\)
\(L(\frac12)\) \(\approx\) \(2.47232 + 1.51493i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.39e5 + 2.74e5i)T \)
good2 \( 1 - 16.0iT - 128T^{2} \)
3 \( 1 - 52.6T + 2.18e3T^{2} \)
5 \( 1 + 434. iT - 7.81e4T^{2} \)
7 \( 1 - 1.47e3T + 8.23e5T^{2} \)
11 \( 1 - 4.55e3T + 1.94e7T^{2} \)
13 \( 1 + 4.82e3iT - 6.27e7T^{2} \)
17 \( 1 - 3.56e3iT - 4.10e8T^{2} \)
19 \( 1 - 1.90e4iT - 8.93e8T^{2} \)
23 \( 1 - 8.26e3iT - 3.40e9T^{2} \)
29 \( 1 - 1.53e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.12e5iT - 2.75e10T^{2} \)
41 \( 1 - 1.20e5T + 1.94e11T^{2} \)
43 \( 1 - 7.07e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.36e6T + 5.06e11T^{2} \)
53 \( 1 - 1.19e6T + 1.17e12T^{2} \)
59 \( 1 - 1.74e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.21e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.96e6T + 6.06e12T^{2} \)
71 \( 1 + 4.60e6T + 9.09e12T^{2} \)
73 \( 1 + 3.25e6T + 1.10e13T^{2} \)
79 \( 1 + 8.69e6iT - 1.92e13T^{2} \)
83 \( 1 + 1.02e7T + 2.71e13T^{2} \)
89 \( 1 - 6.42e5iT - 4.42e13T^{2} \)
97 \( 1 - 7.33e6iT - 8.07e13T^{2} \)
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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

−14.75519647303485491381726830865, −14.46922352607617731246707945321, −13.17049023164037888190593957575, −11.59927834943557056817062182552, −9.135347875994776303758743008545, −8.395890787696997345737127753208, −7.73636271354534304466868555598, −5.58142162957024878972755417170, −4.39430890316677502417753755674, −1.53626374310236814417939768956, 1.75340019821923220130719309388, 2.78751676481750629709516192414, 4.10363324004686244177489515352, 6.99101637163987423197210234436, 8.532930558904771667508329135621, 9.863847700456643686886909898593, 11.22411834232565101773019611809, 11.63675325063509644021438248112, 13.71687829908822200539764648092, 14.36564281821951927804035049662

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