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Properties

Label 2-1932-1932.335-c0-0-0
Degree $2$
Conductor $1932$
Sign $-0.763 - 0.645i$
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  + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.118 + 0.258i)5-s + (−0.142 − 0.989i)6-s + (0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.273 + 0.0801i)10-s + (−1.25 + 1.45i)11-s + (0.654 − 0.755i)12-s + (0.415 + 0.909i)14-s + (0.239 − 0.153i)15-s + (−0.959 − 0.281i)16-s + (−0.239 − 1.66i)17-s + (−0.415 + 0.909i)18-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.118 + 0.258i)5-s + (−0.142 − 0.989i)6-s + (0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.273 + 0.0801i)10-s + (−1.25 + 1.45i)11-s + (0.654 − 0.755i)12-s + (0.415 + 0.909i)14-s + (0.239 − 0.153i)15-s + (−0.959 − 0.281i)16-s + (−0.239 − 1.66i)17-s + (−0.415 + 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.763 - 0.645i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ -0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001379194\)
\(L(\frac12)\) \(\approx\) \(1.001379194\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
good5 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.959 - 0.281i)T^{2} \)
31 \( 1 + (1.41 - 0.909i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.415 - 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 - 0.755i)T^{2} \)
89 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.654 + 0.755i)T^{2} \)
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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

−9.705645718774634581746561799344, −8.521035436189221334870106810489, −7.75266275051361210327331240813, −7.27373836717116933154914683026, −6.64446835132714426963355210075, −5.49703837372939437786513030138, −4.99922805380074213115686757371, −4.47856191600261602555843662752, −2.88454489828182226567789843932, −1.91097241347620836747954136895, 0.64200916317870624271037496418, 2.06539921458736989228957734452, 3.42268079130048139270434050060, 4.15415337311112986758955637273, 5.05034204331768559820335133215, 5.55747054506900395632972309573, 6.28070394500328089411516507199, 7.48047479166656173920132251048, 8.552186335264053887597417562780, 9.139872507988512009509524198376

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