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Properties

Label 2-1932-1932.1595-c0-0-1
Degree $2$
Conductor $1932$
Sign $0.969 + 0.246i$
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  + (0.142 − 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−1.25 − 1.45i)5-s + (0.959 − 0.281i)6-s + (0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−1.61 + 1.03i)10-s + (0.239 + 1.66i)11-s + (−0.142 − 0.989i)12-s + (0.654 − 0.755i)14-s + (0.797 − 1.74i)15-s + (0.841 + 0.540i)16-s + (0.797 − 0.234i)17-s + (0.654 + 0.755i)18-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−1.25 − 1.45i)5-s + (0.959 − 0.281i)6-s + (0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−1.61 + 1.03i)10-s + (0.239 + 1.66i)11-s + (−0.142 − 0.989i)12-s + (0.654 − 0.755i)14-s + (0.797 − 1.74i)15-s + (0.841 + 0.540i)16-s + (0.797 − 0.234i)17-s + (0.654 + 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (1595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104340553\)
\(L(\frac12)\) \(\approx\) \(1.104340553\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
good5 \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \)
11 \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.797 + 0.234i)T + (0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
29 \( 1 + (-0.841 + 0.540i)T^{2} \)
31 \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
41 \( 1 + (0.186 + 0.215i)T + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.654 - 0.755i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.142 - 0.989i)T^{2} \)
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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

−9.301893390425514101130029790471, −8.882397715098806247395285482223, −7.986565494573867216656581293905, −7.57430721655697873719928075707, −5.41690361171246483815789144051, −5.00518601426585264682060840640, −4.32817943793984614301629300521, −3.78563337973708479656685934062, −2.50043505712256817471065449189, −1.30926360080843371259700266729, 0.904269829482794835061080420072, 3.09146771837982004352052339695, 3.37120473527418301267167256795, 4.45935722555444074864234191139, 5.74386795593499721048573103704, 6.56943295467814901697476313511, 7.07873645473338848571393052301, 7.898455115978111465244025334474, 8.138042757509771856158926086617, 8.901367089689085293832778989419

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