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Properties

Label 2-3204-1068.191-c0-0-1
Degree $2$
Conductor $3204$
Sign $0.962 - 0.272i$
Analytic cond. $1.59900$
Root an. cond. $1.26451$
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  + (−0.540 + 0.841i)2-s + (−0.415 − 0.909i)4-s + (1.21 − 0.905i)5-s + (0.989 + 0.142i)8-s + (0.107 + 1.50i)10-s + (1.89 + 0.0677i)13-s + (−0.654 + 0.755i)16-s + (0.936 + 0.203i)17-s + (−1.32 − 0.724i)20-s + (0.361 − 1.23i)25-s + (−1.08 + 1.55i)26-s + (−1.21 + 0.310i)29-s + (−0.281 − 0.959i)32-s + (−0.677 + 0.677i)34-s + (1.05 − 0.436i)37-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + (−0.415 − 0.909i)4-s + (1.21 − 0.905i)5-s + (0.989 + 0.142i)8-s + (0.107 + 1.50i)10-s + (1.89 + 0.0677i)13-s + (−0.654 + 0.755i)16-s + (0.936 + 0.203i)17-s + (−1.32 − 0.724i)20-s + (0.361 − 1.23i)25-s + (−1.08 + 1.55i)26-s + (−1.21 + 0.310i)29-s + (−0.281 − 0.959i)32-s + (−0.677 + 0.677i)34-s + (1.05 − 0.436i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.962 - 0.272i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.962 - 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283055573\)
\(L(\frac12)\) \(\approx\) \(1.283055573\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.540 - 0.841i)T \)
3 \( 1 \)
89 \( 1 + (-0.599 + 0.800i)T \)
good5 \( 1 + (-1.21 + 0.905i)T + (0.281 - 0.959i)T^{2} \)
7 \( 1 + (-0.479 - 0.877i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-1.89 - 0.0677i)T + (0.997 + 0.0713i)T^{2} \)
17 \( 1 + (-0.936 - 0.203i)T + (0.909 + 0.415i)T^{2} \)
19 \( 1 + (-0.599 - 0.800i)T^{2} \)
23 \( 1 + (0.800 - 0.599i)T^{2} \)
29 \( 1 + (1.21 - 0.310i)T + (0.877 - 0.479i)T^{2} \)
31 \( 1 + (-0.800 - 0.599i)T^{2} \)
37 \( 1 + (-1.05 + 0.436i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.99 - 0.0713i)T + (0.997 - 0.0713i)T^{2} \)
43 \( 1 + (0.877 + 0.479i)T^{2} \)
47 \( 1 + (0.755 + 0.654i)T^{2} \)
53 \( 1 + (1.83 - 0.682i)T + (0.755 - 0.654i)T^{2} \)
59 \( 1 + (0.0713 + 0.997i)T^{2} \)
61 \( 1 + (1.71 + 0.183i)T + (0.977 + 0.212i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.281 + 0.959i)T^{2} \)
73 \( 1 + (-0.627 - 0.544i)T + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.989 + 0.142i)T^{2} \)
83 \( 1 + (0.936 - 0.349i)T^{2} \)
97 \( 1 + (-0.0994 + 0.691i)T + (-0.959 - 0.281i)T^{2} \)
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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

−8.954252294562935217459752997257, −8.220527429935875091297828248983, −7.53206229084970645703842019323, −6.35742938664831660101325932226, −5.98428425227693204130792005725, −5.37210364478285449101694012159, −4.52348355096744043709023983768, −3.43356976009834068644800853696, −1.76227713074234695945883974057, −1.20197151386702051531472656938, 1.32920294184181421757266985595, 2.09735066693577889324310119945, 3.21903473534475989799460423526, 3.62800786538496993402275152681, 4.96432584139771839398799298462, 5.95415210220568173441165951055, 6.48545907010600294446342421777, 7.47004876101171996798102162480, 8.229098141982529418391870053595, 8.994311493206904932373030811726

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