Properties

Label 2-37-37.8-c4-0-6
Degree $2$
Conductor $37$
Sign $0.766 - 0.642i$
Analytic cond. $3.82468$
Root an. cond. $1.95568$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.54 + 1.48i)2-s + (7.08 + 4.09i)3-s + (14.6 + 8.46i)4-s + (−12.9 + 3.45i)5-s + (33.2 + 33.2i)6-s + (8.16 − 14.1i)7-s + (3.74 + 3.74i)8-s + (−7.01 − 12.1i)9-s − 76.6·10-s + 10.9i·11-s + (69.2 + 119. i)12-s + (89.0 − 23.8i)13-s + (66.2 − 66.2i)14-s + (−105. − 28.2i)15-s + (−120. − 208. i)16-s + (−64.1 + 239. i)17-s + ⋯
L(s)  = 1  + (1.38 + 0.371i)2-s + (0.787 + 0.454i)3-s + (0.915 + 0.528i)4-s + (−0.516 + 0.138i)5-s + (0.922 + 0.922i)6-s + (0.166 − 0.288i)7-s + (0.0584 + 0.0584i)8-s + (−0.0865 − 0.149i)9-s − 0.766·10-s + 0.0902i·11-s + (0.480 + 0.832i)12-s + (0.526 − 0.141i)13-s + (0.337 − 0.337i)14-s + (−0.469 − 0.125i)15-s + (−0.469 − 0.813i)16-s + (−0.221 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(3.82468\)
Root analytic conductor: \(1.95568\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :2),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.91278 + 1.05945i\)
\(L(\frac12)\) \(\approx\) \(2.91278 + 1.05945i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.16e3 - 711. i)T \)
good2 \( 1 + (-5.54 - 1.48i)T + (13.8 + 8i)T^{2} \)
3 \( 1 + (-7.08 - 4.09i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (12.9 - 3.45i)T + (541. - 312.5i)T^{2} \)
7 \( 1 + (-8.16 + 14.1i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 - 10.9iT - 1.46e4T^{2} \)
13 \( 1 + (-89.0 + 23.8i)T + (2.47e4 - 1.42e4i)T^{2} \)
17 \( 1 + (64.1 - 239. i)T + (-7.23e4 - 4.17e4i)T^{2} \)
19 \( 1 + (-161. + 43.2i)T + (1.12e5 - 6.51e4i)T^{2} \)
23 \( 1 + (121. + 121. i)T + 2.79e5iT^{2} \)
29 \( 1 + (454. - 454. i)T - 7.07e5iT^{2} \)
31 \( 1 + (356. - 356. i)T - 9.23e5iT^{2} \)
41 \( 1 + (-1.48e3 - 857. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-2.13e3 - 2.13e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 756.T + 4.87e6T^{2} \)
53 \( 1 + (553. + 959. i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-162. + 604. i)T + (-1.04e7 - 6.05e6i)T^{2} \)
61 \( 1 + (1.02e3 + 3.83e3i)T + (-1.19e7 + 6.92e6i)T^{2} \)
67 \( 1 + (6.59e3 + 3.80e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (1.36e3 - 2.36e3i)T + (-1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + 5.95e3iT - 2.83e7T^{2} \)
79 \( 1 + (-1.08e4 + 2.91e3i)T + (3.37e7 - 1.94e7i)T^{2} \)
83 \( 1 + (-2.21e3 - 3.83e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (2.76e3 + 740. i)T + (5.43e7 + 3.13e7i)T^{2} \)
97 \( 1 + (9.55e3 + 9.55e3i)T + 8.85e7iT^{2} \)
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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

−15.29552056046865343241895019702, −14.63266595595554212712465745454, −13.66837455518856399988278092509, −12.51144536478318719080179540409, −11.14550163448450894766592764024, −9.356868843271985352049030091229, −7.79569408932709233841553978864, −6.12749125330289130704411061396, −4.33185444538554468883845729605, −3.28050206707961439842509012883, 2.42769960754256912862568848452, 3.97675023616126006633413079213, 5.64474593557077079770389951999, 7.60761679375545678702349687043, 8.956688957905915031258380249801, 11.16374124553241497490734043293, 12.11278086674936113507739959059, 13.34112534822305120361596315688, 13.99905077287157567278599142004, 15.05948480665688280180353721602

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