Properties

Label 2-37-37.36-c3-0-6
Degree $2$
Conductor $37$
Sign $-0.916 + 0.399i$
Analytic cond. $2.18307$
Root an. cond. $1.47752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.39i·2-s + 2.81·3-s − 21.1·4-s − 6.77i·5-s − 15.1i·6-s + 16.2·7-s + 70.7i·8-s − 19.0·9-s − 36.5·10-s + 56.6·11-s − 59.4·12-s − 41.6i·13-s − 87.7i·14-s − 19.0i·15-s + 212.·16-s + 68.5i·17-s + ⋯
L(s)  = 1  − 1.90i·2-s + 0.541·3-s − 2.63·4-s − 0.606i·5-s − 1.03i·6-s + 0.877·7-s + 3.12i·8-s − 0.706·9-s − 1.15·10-s + 1.55·11-s − 1.43·12-s − 0.888i·13-s − 1.67i·14-s − 0.328i·15-s + 3.32·16-s + 0.977i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.916 + 0.399i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ -0.916 + 0.399i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.270609 - 1.29889i\)
\(L(\frac12)\) \(\approx\) \(0.270609 - 1.29889i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (206. - 89.8i)T \)
good2 \( 1 + 5.39iT - 8T^{2} \)
3 \( 1 - 2.81T + 27T^{2} \)
5 \( 1 + 6.77iT - 125T^{2} \)
7 \( 1 - 16.2T + 343T^{2} \)
11 \( 1 - 56.6T + 1.33e3T^{2} \)
13 \( 1 + 41.6iT - 2.19e3T^{2} \)
17 \( 1 - 68.5iT - 4.91e3T^{2} \)
19 \( 1 + 84.5iT - 6.85e3T^{2} \)
23 \( 1 - 70.4iT - 1.21e4T^{2} \)
29 \( 1 - 111. iT - 2.43e4T^{2} \)
31 \( 1 - 322. iT - 2.97e4T^{2} \)
41 \( 1 - 131.T + 6.89e4T^{2} \)
43 \( 1 + 112. iT - 7.95e4T^{2} \)
47 \( 1 + 98.6T + 1.03e5T^{2} \)
53 \( 1 - 215.T + 1.48e5T^{2} \)
59 \( 1 + 40.1iT - 2.05e5T^{2} \)
61 \( 1 + 720. iT - 2.26e5T^{2} \)
67 \( 1 + 476.T + 3.00e5T^{2} \)
71 \( 1 + 664.T + 3.57e5T^{2} \)
73 \( 1 - 281.T + 3.89e5T^{2} \)
79 \( 1 + 706. iT - 4.93e5T^{2} \)
83 \( 1 + 370.T + 5.71e5T^{2} \)
89 \( 1 - 818. iT - 7.04e5T^{2} \)
97 \( 1 - 63.4iT - 9.12e5T^{2} \)
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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

−14.74427829605350781919074018896, −13.91990212907247305978799102254, −12.68191394123175853599222571520, −11.69057350094681813747364378603, −10.67738040920029770345049469035, −9.065632784024298439309767627271, −8.494506591339093111099969573045, −4.98234314011458615500957577835, −3.39205103370227674598536145932, −1.42947823421608526138806451791, 4.22204746749457819931203692813, 6.02807938425407256996016691887, 7.25307284704792828827910614884, 8.478552240628202469553752035736, 9.437984373326311268628279652721, 11.66866378929365963643004046299, 13.77621698555938224951671436187, 14.48199110544285619435665847305, 14.83746078056025791132702893919, 16.46666734377506745475136447377

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