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Properties

Label 2-1932-1932.167-c0-0-3
Degree $2$
Conductor $1932$
Sign $-0.117 + 0.993i$
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  + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (1.10 − 0.708i)5-s + (0.654 − 0.755i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.186 − 1.29i)10-s + (−0.118 − 0.258i)11-s + (−0.415 − 0.909i)12-s + (−0.841 − 0.540i)14-s + (1.25 − 0.368i)15-s + (−0.142 + 0.989i)16-s + (−1.25 + 1.45i)17-s + (0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (1.10 − 0.708i)5-s + (0.654 − 0.755i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.186 − 1.29i)10-s + (−0.118 − 0.258i)11-s + (−0.415 − 0.909i)12-s + (−0.841 − 0.540i)14-s + (1.25 − 0.368i)15-s + (−0.142 + 0.989i)16-s + (−1.25 + 1.45i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(0.964193\)
Root analytic conductor: \(0.981933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :0),\ -0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.112328729\)
\(L(\frac12)\) \(\approx\) \(2.112328729\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
good5 \( 1 + (-1.10 + 0.708i)T + (0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
41 \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (-1.61 - 0.474i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−9.368889122165463683239485287298, −8.597725983076813608724892941715, −7.986926279272560512253211101158, −6.65026136890280586278923527408, −5.78003021113198530851481945094, −4.78882026670887337367460072367, −4.13969748953439048739532360813, −3.32387415225279852778854160801, −2.05263268724437834789322140143, −1.47338853352392944579465701523, 2.20398432568131238282558154244, 2.64013465187719775986658860493, 3.76555693902691296895125295056, 4.94970938593185835715944402420, 5.69772764890637149426236679725, 6.52995040531672479856223990811, 7.19986246929103049041459139592, 7.83195831438490954852571244154, 8.987606012118994987302062686653, 9.262227034954329947839882287766

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