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Properties

Label 2-3204-1068.1055-c0-0-0
Degree $2$
Conductor $3204$
Sign $-0.845 - 0.534i$
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 + (0.0225 − 0.631i)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 + (0.518 + 0.360i)26-s + (0.385 + 1.50i)29-s + (−0.281 − 0.959i)32-s + (−0.677 + 0.677i)34-s + (0.628 + 1.51i)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 + (0.0225 − 0.631i)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 + (0.518 + 0.360i)26-s + (0.385 + 1.50i)29-s + (−0.281 − 0.959i)32-s + (−0.677 + 0.677i)34-s + (0.628 + 1.51i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $-0.845 - 0.534i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (2123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ -0.845 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5645270181\)
\(L(\frac12)\) \(\approx\) \(0.5645270181\)
\(L(1)\) 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 \( 1 + (-0.0225 + 0.631i)T + (-0.997 - 0.0713i)T^{2} \)
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 \( 1 + (-0.385 - 1.50i)T + (-0.877 + 0.479i)T^{2} \)
31 \( 1 + (0.800 + 0.599i)T^{2} \)
37 \( 1 + (-0.628 - 1.51i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.00254 + 0.0713i)T + (-0.997 + 0.0713i)T^{2} \)
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 \( 1 + (0.109 - 1.01i)T + (-0.977 - 0.212i)T^{2} \)
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

−8.943286284035306336198867464638, −8.040893055184638956626872974389, −7.84248886892489669758385442124, −6.96899816756346035200969293813, −6.44287694571374669645166294148, −5.46734087189579510943262965753, −4.66570303200344271936288700656, −3.65005191922906749186566930472, −2.89372223752874500416602697498, −1.22746200259576627586866824440, 0.48157666415684443214411111866, 1.65442390296989517587405853635, 2.89050305593713551064650706500, 3.85360762033638971810553508129, 4.37304452186302592562652739767, 5.16811017249536072963132374656, 6.39934698240269234315840691794, 7.57352726345882869947584694448, 7.86348444834286203579366127050, 8.551964614223634482525558584641

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