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Properties

Label 2-3204-1068.155-c0-0-0
Degree $2$
Conductor $3204$
Sign $0.891 - 0.453i$
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  + (0.909 − 0.415i)2-s + (0.654 − 0.755i)4-s + (−1.73 + 0.948i)5-s + (0.281 − 0.959i)8-s + (−1.18 + 1.58i)10-s + (0.156 + 0.469i)13-s + (−0.142 − 0.989i)16-s + (0.682 + 1.83i)17-s + (−0.420 + 1.93i)20-s + (1.57 − 2.45i)25-s + (0.337 + 0.362i)26-s + (1.53 + 1.23i)29-s + (−0.540 − 0.841i)32-s + (1.38 + 1.38i)34-s + (−0.521 + 1.25i)37-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)2-s + (0.654 − 0.755i)4-s + (−1.73 + 0.948i)5-s + (0.281 − 0.959i)8-s + (−1.18 + 1.58i)10-s + (0.156 + 0.469i)13-s + (−0.142 − 0.989i)16-s + (0.682 + 1.83i)17-s + (−0.420 + 1.93i)20-s + (1.57 − 2.45i)25-s + (0.337 + 0.362i)26-s + (1.53 + 1.23i)29-s + (−0.540 − 0.841i)32-s + (1.38 + 1.38i)34-s + (−0.521 + 1.25i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.891 - 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.566908767\)
\(L(\frac12)\) \(\approx\) \(1.566908767\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 + 0.415i)T \)
3 \( 1 \)
89 \( 1 + (-0.479 + 0.877i)T \)
good5 \( 1 + (1.73 - 0.948i)T + (0.540 - 0.841i)T^{2} \)
7 \( 1 + (-0.977 + 0.212i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.156 - 0.469i)T + (-0.800 + 0.599i)T^{2} \)
17 \( 1 + (-0.682 - 1.83i)T + (-0.755 + 0.654i)T^{2} \)
19 \( 1 + (-0.479 - 0.877i)T^{2} \)
23 \( 1 + (-0.877 + 0.479i)T^{2} \)
29 \( 1 + (-1.53 - 1.23i)T + (0.212 + 0.977i)T^{2} \)
31 \( 1 + (0.877 + 0.479i)T^{2} \)
37 \( 1 + (0.521 - 1.25i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.199 - 0.599i)T + (-0.800 - 0.599i)T^{2} \)
43 \( 1 + (0.212 - 0.977i)T^{2} \)
47 \( 1 + (0.989 - 0.142i)T^{2} \)
53 \( 1 + (0.697 + 0.0498i)T + (0.989 + 0.142i)T^{2} \)
59 \( 1 + (-0.599 + 0.800i)T^{2} \)
61 \( 1 + (-1.63 - 1.13i)T + (0.349 + 0.936i)T^{2} \)
67 \( 1 + (0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.540 + 0.841i)T^{2} \)
73 \( 1 + (1.29 - 0.186i)T + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.281 - 0.959i)T^{2} \)
83 \( 1 + (0.997 + 0.0713i)T^{2} \)
97 \( 1 + (-0.136 - 0.0401i)T + (0.841 + 0.540i)T^{2} \)
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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

−8.616627152724284047210066941266, −8.138701899450947182024265229914, −7.15273375809045873906345184735, −6.70987194564085745461785631361, −5.92323436982774293930198857458, −4.74956029602286537791805831118, −4.08766516558694953942576468059, −3.43721401678607730022086587386, −2.80647505233360536920529339297, −1.37481286617886814875699030058, 0.77982380407643884635846703142, 2.62511907782419859564090393535, 3.51049327773065550805149891319, 4.16996085344610707893425132369, 4.96192821470580913234157652038, 5.41315409303762895191133242475, 6.62399233865640384267187333210, 7.46236492893711819479818854594, 7.79703284774437932205807854633, 8.529643119091996356742520929468

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