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Properties

Label 2-1932-1932.587-c0-0-1
Degree $2$
Conductor $1932$
Sign $0.451 - 0.892i$
Analytic cond. $0.964193$
Root an. cond. $0.981933$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (0.797 − 0.234i)5-s + (−0.415 − 0.909i)6-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.544 + 0.627i)10-s + (1.10 + 0.708i)11-s + (0.841 + 0.540i)12-s + (0.959 + 0.281i)14-s + (0.118 + 0.822i)15-s + (−0.654 − 0.755i)16-s + (0.118 + 0.258i)17-s + (0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (0.797 − 0.234i)5-s + (−0.415 − 0.909i)6-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.544 + 0.627i)10-s + (1.10 + 0.708i)11-s + (0.841 + 0.540i)12-s + (0.959 + 0.281i)14-s + (0.118 + 0.822i)15-s + (−0.654 − 0.755i)16-s + (0.118 + 0.258i)17-s + (0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ 0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8315452671\)
\(L(\frac12)\) \(\approx\) \(0.8315452671\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.415 - 0.909i)T \)
good5 \( 1 + (-0.797 + 0.234i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.654 - 0.755i)T^{2} \)
31 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
37 \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.959 + 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.273 - 1.89i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−9.418940751061957591447121998034, −9.295589176269763876486773008877, −8.023694935598836010965033192160, −7.11362774917218831591567997384, −6.38461068971317652626152061114, −5.68334125465438319799432258088, −4.77230547769784970157601655131, −3.88524320982884505809294824215, −2.58163261470369493850481160404, −1.12325817816109818516164024613, 1.06415292633846106951842680058, 2.17584526383726593858372646319, 2.88397305150120862235149302742, 3.95262113205066515837825680182, 5.86640758402313288788516120100, 6.05882933421651606337272654538, 6.88025739454192603429209951669, 7.893562703653225983555446600606, 8.456554260829189064559180494183, 9.462430643115743162177662830685

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