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Properties

Label 2-936-104.11-c1-0-2
Degree $2$
Conductor $936$
Sign $-0.336 + 0.941i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.563 + 1.29i)2-s + (−1.36 + 1.46i)4-s + (0.00433 − 0.00433i)5-s + (−1.57 + 0.421i)7-s + (−2.66 − 0.946i)8-s + (0.00805 + 0.00317i)10-s + (−3.23 − 0.865i)11-s + (3.04 + 1.92i)13-s + (−1.43 − 1.80i)14-s + (−0.274 − 3.99i)16-s + (−4.80 − 2.77i)17-s + (−3.45 + 0.926i)19-s + (0.000419 + 0.0122i)20-s + (−0.697 − 4.67i)22-s + (−2.76 − 4.78i)23-s + ⋯
L(s)  = 1  + (0.398 + 0.917i)2-s + (−0.682 + 0.730i)4-s + (0.00193 − 0.00193i)5-s + (−0.594 + 0.159i)7-s + (−0.942 − 0.334i)8-s + (0.00254 + 0.00100i)10-s + (−0.974 − 0.261i)11-s + (0.844 + 0.535i)13-s + (−0.383 − 0.482i)14-s + (−0.0685 − 0.997i)16-s + (−1.16 − 0.673i)17-s + (−0.793 + 0.212i)19-s + (9.39e−5 + 0.00273i)20-s + (−0.148 − 0.997i)22-s + (−0.575 − 0.997i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0642520 - 0.0912418i\)
\(L(\frac12)\) \(\approx\) \(0.0642520 - 0.0912418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.563 - 1.29i)T \)
3 \( 1 \)
13 \( 1 + (-3.04 - 1.92i)T \)
good5 \( 1 + (-0.00433 + 0.00433i)T - 5iT^{2} \)
7 \( 1 + (1.57 - 0.421i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.23 + 0.865i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (4.80 + 2.77i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 - 0.926i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.76 + 4.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.51 - 0.876i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.487 - 0.487i)T - 31iT^{2} \)
37 \( 1 + (-3.01 + 11.2i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.29 - 4.82i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.33 + 5.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.87 - 2.87i)T + 47iT^{2} \)
53 \( 1 - 3.71iT - 53T^{2} \)
59 \( 1 + (0.772 + 2.88i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.81 - 1.04i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.133 + 0.497i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.569 + 2.12i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - 73iT^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 + (-3.55 - 3.55i)T + 83iT^{2} \)
89 \( 1 + (14.4 + 3.88i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.70 + 1.26i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.67028669685789397072989831718, −9.504949847913898463001296608974, −8.824928468811037783416033441633, −8.109702410609640915847149527243, −7.07512701883180130368883363650, −6.37951523835156717370211553540, −5.58914085457277910484243088871, −4.56055408622704654448916411989, −3.63401428690982226066334020769, −2.44745826977782557843165361312, 0.04287431628213049675620193516, 1.83633314469463666927806691783, 2.95314221969829350703348215345, 3.92848716089463517723048294647, 4.84729848139332354174327652416, 5.93469139802980646275804417728, 6.62007079040945004741341134447, 8.097269814878303135195323504255, 8.693338667720620587802249874059, 9.858599504404019236740760647213

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