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Properties

Label 2-3204-1068.239-c0-0-1
Degree $2$
Conductor $3204$
Sign $0.656 - 0.754i$
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  + (0.989 + 0.142i)2-s + (0.959 + 0.281i)4-s + (0.377 + 1.01i)5-s + (0.909 + 0.415i)8-s + (0.229 + 1.05i)10-s + (1.13 + 0.121i)13-s + (0.841 + 0.540i)16-s + (−1.59 − 1.19i)17-s + (0.0771 + 1.07i)20-s + (−0.127 + 0.110i)25-s + (1.10 + 0.282i)26-s + (1.30 − 1.21i)29-s + (0.755 + 0.654i)32-s + (−1.41 − 1.41i)34-s + (−0.741 + 1.79i)37-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)2-s + (0.959 + 0.281i)4-s + (0.377 + 1.01i)5-s + (0.909 + 0.415i)8-s + (0.229 + 1.05i)10-s + (1.13 + 0.121i)13-s + (0.841 + 0.540i)16-s + (−1.59 − 1.19i)17-s + (0.0771 + 1.07i)20-s + (−0.127 + 0.110i)25-s + (1.10 + 0.282i)26-s + (1.30 − 1.21i)29-s + (0.755 + 0.654i)32-s + (−1.41 − 1.41i)34-s + (−0.741 + 1.79i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (2375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.656 - 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.622349309\)
\(L(\frac12)\) \(\approx\) \(2.622349309\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.989 - 0.142i)T \)
3 \( 1 \)
89 \( 1 + (0.936 + 0.349i)T \)
good5 \( 1 + (-0.377 - 1.01i)T + (-0.755 + 0.654i)T^{2} \)
7 \( 1 + (0.997 + 0.0713i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (-1.13 - 0.121i)T + (0.977 + 0.212i)T^{2} \)
17 \( 1 + (1.59 + 1.19i)T + (0.281 + 0.959i)T^{2} \)
19 \( 1 + (0.936 - 0.349i)T^{2} \)
23 \( 1 + (-0.349 - 0.936i)T^{2} \)
29 \( 1 + (-1.30 + 1.21i)T + (0.0713 - 0.997i)T^{2} \)
31 \( 1 + (0.349 - 0.936i)T^{2} \)
37 \( 1 + (0.741 - 1.79i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.97 - 0.212i)T + (0.977 - 0.212i)T^{2} \)
43 \( 1 + (0.0713 + 0.997i)T^{2} \)
47 \( 1 + (-0.540 + 0.841i)T^{2} \)
53 \( 1 + (0.767 - 1.40i)T + (-0.540 - 0.841i)T^{2} \)
59 \( 1 + (-0.212 - 0.977i)T^{2} \)
61 \( 1 + (-0.0677 - 0.0225i)T + (0.800 + 0.599i)T^{2} \)
67 \( 1 + (-0.841 + 0.540i)T^{2} \)
71 \( 1 + (-0.755 - 0.654i)T^{2} \)
73 \( 1 + (-1.03 + 1.61i)T + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (-0.909 + 0.415i)T^{2} \)
83 \( 1 + (0.479 - 0.877i)T^{2} \)
97 \( 1 + (-0.729 + 1.59i)T + (-0.654 - 0.755i)T^{2} \)
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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

−8.778830552485372360531097190012, −8.109643793958494920158163698555, −7.07195673330308233999186064444, −6.47863141963008268744527075530, −6.22852478755438783714257535411, −4.98180874363035443969007322429, −4.41949954942193969472525407136, −3.27367173482710813649469953834, −2.74693933193050842855683517110, −1.71649966129947441900478499303, 1.34086113802691140223071885695, 2.08723238721857056307572801535, 3.39915637752355045185733903583, 4.08670146682800421259238932087, 4.97204931827544344097528261672, 5.48175507278791252369117990051, 6.49965926044106018181790190230, 6.79675905500059617394835333549, 8.220844450519157818582027371702, 8.595847659844009611095351167872

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