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Properties

Label 2-1932-1932.1007-c0-0-1
Degree $2$
Conductor $1932$
Sign $0.899 - 0.436i$
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  + (0.959 + 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.239 + 1.66i)5-s + (0.841 − 0.540i)6-s + (−0.415 − 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.698 + 1.53i)10-s + (0.797 − 0.234i)11-s + (0.959 − 0.281i)12-s + (−0.142 − 0.989i)14-s + (1.10 + 1.27i)15-s + (0.415 + 0.909i)16-s + (−1.10 + 0.708i)17-s + (0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.239 + 1.66i)5-s + (0.841 − 0.540i)6-s + (−0.415 − 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.698 + 1.53i)10-s + (0.797 − 0.234i)11-s + (0.959 − 0.281i)12-s + (−0.142 − 0.989i)14-s + (1.10 + 1.27i)15-s + (0.415 + 0.909i)16-s + (−1.10 + 0.708i)17-s + (0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ 0.899 - 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.426096479\)
\(L(\frac12)\) \(\approx\) \(2.426096479\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 - 0.281i)T \)
3 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
good5 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
11 \( 1 + (-0.797 + 0.234i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (0.142 + 0.989i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−9.470133052244926898953748698345, −8.251655619785607267814664208346, −7.48613442124344007184268597204, −7.07707067892274322839649776905, −6.38909862895106564264112630417, −5.85677358363489943753683065479, −3.88924791415565875961532427399, −3.80793851890475388806004075230, −2.83487537634756282271099146494, −1.83802455203083398186370018613, 1.51111361499795759431557389857, 2.66717758954498550760027761430, 3.60856919361368902409625844569, 4.51956886007885120648254258515, 5.03290848130050920265826495513, 5.68890246994188827234388405335, 6.88267632305253501711185206000, 7.87456348994135362370308265821, 8.852020966767741943117575831267, 9.396987046714650999769045374716

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