Properties

Label 2-37-37.36-c7-0-11
Degree $2$
Conductor $37$
Sign $0.578 - 0.815i$
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  + 4.18i·2-s + 63.9·3-s + 110.·4-s + 270. i·5-s + 267. i·6-s + 565.·7-s + 998. i·8-s + 1.90e3·9-s − 1.13e3·10-s − 4.24e3·11-s + 7.06e3·12-s − 1.35e4i·13-s + 2.36e3i·14-s + 1.72e4i·15-s + 9.96e3·16-s + 1.40e4i·17-s + ⋯
L(s)  = 1  + 0.370i·2-s + 1.36·3-s + 0.863·4-s + 0.967i·5-s + 0.506i·6-s + 0.623·7-s + 0.689i·8-s + 0.870·9-s − 0.357·10-s − 0.961·11-s + 1.18·12-s − 1.71i·13-s + 0.230i·14-s + 1.32i·15-s + 0.608·16-s + 0.695i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.578 - 0.815i$
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: \((2,\ 37,\ (\ :7/2),\ 0.578 - 0.815i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.93020 + 1.51349i\)
\(L(\frac12)\) \(\approx\) \(2.93020 + 1.51349i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.78e5 + 2.51e5i)T \)
good2 \( 1 - 4.18iT - 128T^{2} \)
3 \( 1 - 63.9T + 2.18e3T^{2} \)
5 \( 1 - 270. iT - 7.81e4T^{2} \)
7 \( 1 - 565.T + 8.23e5T^{2} \)
11 \( 1 + 4.24e3T + 1.94e7T^{2} \)
13 \( 1 + 1.35e4iT - 6.27e7T^{2} \)
17 \( 1 - 1.40e4iT - 4.10e8T^{2} \)
19 \( 1 - 2.43e4iT - 8.93e8T^{2} \)
23 \( 1 - 2.87e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.59e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.56e5iT - 2.75e10T^{2} \)
41 \( 1 + 3.75e4T + 1.94e11T^{2} \)
43 \( 1 - 4.26e3iT - 2.71e11T^{2} \)
47 \( 1 - 4.51e5T + 5.06e11T^{2} \)
53 \( 1 + 1.43e6T + 1.17e12T^{2} \)
59 \( 1 + 2.76e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.41e6iT - 3.14e12T^{2} \)
67 \( 1 + 3.53e6T + 6.06e12T^{2} \)
71 \( 1 - 6.72e5T + 9.09e12T^{2} \)
73 \( 1 - 5.22e6T + 1.10e13T^{2} \)
79 \( 1 - 2.18e5iT - 1.92e13T^{2} \)
83 \( 1 + 6.56e6T + 2.71e13T^{2} \)
89 \( 1 + 4.16e6iT - 4.42e13T^{2} \)
97 \( 1 + 7.56e6iT - 8.07e13T^{2} \)
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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

−15.05134772851286310458003998636, −14.27992892649053777689662951715, −12.90867756045998796246740038684, −11.10700644124888780494548454938, −10.13557726027437668305921771296, −8.014528383055091904209902161978, −7.72962696530032172444796661528, −5.84485170073968124157825635459, −3.24281598618540462347545610688, −2.22357189971039424985398751398, 1.56059226140633776907760191077, 2.81898674213608796271804244912, 4.70203318386728771625796670730, 7.09174961768016664915210411534, 8.408774250036982369492144715217, 9.390736890313983441717252435357, 11.04270988891925338496504614583, 12.30626103885169041473289615224, 13.53004290512118989562779893772, 14.58854795610156348409930232031

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