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Properties

Label 2-1932-1932.1175-c0-0-3
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: $\chi_{1932} (1175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ 0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6653836972\)
\(L(\frac12)\) \(\approx\) \(0.6653836972\)
\(L(1)\) 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

−9.186487229136748240536733044993, −8.508333164486852020478035716215, −8.154793017425512520135012722391, −7.10497309799990812057274503365, −6.25498113846620107042368999553, −4.72776769997171036295116577673, −4.13325385475860059875435925361, −3.52425202360833294257559823280, −2.29568001663659066506554453176, −0.66712231603625485280581272959, 1.45750846957409575039264290823, 2.17031812835261788144986646835, 3.61735194533846181593000597797, 4.85397240041204111885176868957, 5.99338223716286784905776821455, 6.43420232739003945373851600646, 7.48725736398413234939996453220, 7.84694967239539750154851382320, 8.516276433477629536389553921227, 9.282273075311158295285708221124

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