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Properties

Label 2-3204-1068.83-c0-0-1
Degree $2$
Conductor $3204$
Sign $0.576 - 0.817i$
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  + (−0.281 + 0.959i)2-s + (−0.841 − 0.540i)4-s + (1.81 + 0.129i)5-s + (0.755 − 0.654i)8-s + (−0.635 + 1.70i)10-s + (−0.0702 + 0.0126i)13-s + (0.415 + 0.909i)16-s + (0.574 − 1.05i)17-s + (−1.45 − 1.09i)20-s + (2.28 + 0.328i)25-s + (0.00763 − 0.0709i)26-s + (0.156 + 0.469i)29-s + (−0.989 + 0.142i)32-s + (0.847 + 0.847i)34-s + (−0.0818 + 0.197i)37-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)2-s + (−0.841 − 0.540i)4-s + (1.81 + 0.129i)5-s + (0.755 − 0.654i)8-s + (−0.635 + 1.70i)10-s + (−0.0702 + 0.0126i)13-s + (0.415 + 0.909i)16-s + (0.574 − 1.05i)17-s + (−1.45 − 1.09i)20-s + (2.28 + 0.328i)25-s + (0.00763 − 0.0709i)26-s + (0.156 + 0.469i)29-s + (−0.989 + 0.142i)32-s + (0.847 + 0.847i)34-s + (−0.0818 + 0.197i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.576 - 0.817i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.576 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.437544197\)
\(L(\frac12)\) \(\approx\) \(1.437544197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.281 - 0.959i)T \)
3 \( 1 \)
89 \( 1 + (0.0713 + 0.997i)T \)
good5 \( 1 + (-1.81 - 0.129i)T + (0.989 + 0.142i)T^{2} \)
7 \( 1 + (-0.599 - 0.800i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.0702 - 0.0126i)T + (0.936 - 0.349i)T^{2} \)
17 \( 1 + (-0.574 + 1.05i)T + (-0.540 - 0.841i)T^{2} \)
19 \( 1 + (0.0713 - 0.997i)T^{2} \)
23 \( 1 + (-0.997 - 0.0713i)T^{2} \)
29 \( 1 + (-0.156 - 0.469i)T + (-0.800 + 0.599i)T^{2} \)
31 \( 1 + (0.997 - 0.0713i)T^{2} \)
37 \( 1 + (0.0818 - 0.197i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.93 + 0.349i)T + (0.936 + 0.349i)T^{2} \)
43 \( 1 + (-0.800 - 0.599i)T^{2} \)
47 \( 1 + (-0.909 + 0.415i)T^{2} \)
53 \( 1 + (-0.203 + 0.936i)T + (-0.909 - 0.415i)T^{2} \)
59 \( 1 + (0.349 - 0.936i)T^{2} \)
61 \( 1 + (-1.53 + 0.912i)T + (0.479 - 0.877i)T^{2} \)
67 \( 1 + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (0.989 - 0.142i)T^{2} \)
73 \( 1 + (1.53 - 0.698i)T + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (-0.755 - 0.654i)T^{2} \)
83 \( 1 + (-0.212 + 0.977i)T^{2} \)
97 \( 1 + (-1.27 - 1.47i)T + (-0.142 + 0.989i)T^{2} \)
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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

−8.975311659912762754075348817229, −8.310845103340086910631637784924, −7.21964987144662976465353363312, −6.74323563585352169566591651188, −5.96737216416008114896305910882, −5.30608467340096828274330118770, −4.82721091917954176954408645396, −3.41103700044003089129552714488, −2.26228478476712363064015824886, −1.21037992555163653955631178883, 1.28984466739561828917370623998, 2.01466417436095520156621153291, 2.85641246601996800684689367353, 3.87023761047198842730911125149, 4.93139588570721719203214737038, 5.59665040587064799069716043287, 6.31225208061192158571481192346, 7.31297414272601699933345376485, 8.397190583365843358044147437651, 8.848212896985813688589048462302

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