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Properties

Label 2-804-67.25-c1-0-7
Degree $2$
Conductor $804$
Sign $0.646 + 0.762i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 − 0.755i)3-s + (−3.19 + 2.05i)5-s + (−0.0504 − 0.350i)7-s + (−0.142 − 0.989i)9-s + (0.990 − 0.636i)11-s + (2.04 − 4.47i)13-s + (−0.540 + 3.76i)15-s + (2.93 + 0.861i)17-s + (0.418 − 2.90i)19-s + (−0.298 − 0.191i)21-s + (0.101 − 0.117i)23-s + (3.92 − 8.58i)25-s + (−0.841 − 0.540i)27-s + 10.1·29-s + (−1.39 − 3.06i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (−1.42 + 0.918i)5-s + (−0.0190 − 0.132i)7-s + (−0.0474 − 0.329i)9-s + (0.298 − 0.192i)11-s + (0.567 − 1.24i)13-s + (−0.139 + 0.971i)15-s + (0.711 + 0.208i)17-s + (0.0959 − 0.667i)19-s + (−0.0650 − 0.0418i)21-s + (0.0212 − 0.0244i)23-s + (0.784 − 1.71i)25-s + (−0.161 − 0.104i)27-s + 1.88·29-s + (−0.251 − 0.549i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.646 + 0.762i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.646 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21340 - 0.562040i\)
\(L(\frac12)\) \(\approx\) \(1.21340 - 0.562040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (4.51 + 6.83i)T \)
good5 \( 1 + (3.19 - 2.05i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.0504 + 0.350i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-0.990 + 0.636i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-2.04 + 4.47i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-2.93 - 0.861i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-0.418 + 2.90i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (-0.101 + 0.117i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + (1.39 + 3.06i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 - 3.68T + 37T^{2} \)
41 \( 1 + (-3.40 - 0.998i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-3.71 - 1.09i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-2.23 + 2.57i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (8.63 - 2.53i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (3.22 + 7.06i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (8.93 + 5.74i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (9.64 - 2.83i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (1.96 + 1.26i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (3.74 - 8.19i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-10.9 + 7.02i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (3.15 + 3.63i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 - 1.62T + 97T^{2} \)
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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

−10.41027482634713840361163459890, −9.151546062324642186507158408787, −8.039085718284299890543926047995, −7.80778616720260986511516597441, −6.82286619792546144528181572348, −5.98161657389122402112477207566, −4.48363199208049931247774868184, −3.43194870097443108992795185160, −2.83229192839768700805777655760, −0.76889618699481320807474512988, 1.26861571601196649921896568827, 3.10212727144345123437372519747, 4.18757548885565471697122375431, 4.57304946252065192762788598695, 5.90242976475543670593487198125, 7.17288417194187752081931090399, 7.959733251617932056001440280261, 8.744206557957971842490402288935, 9.253416058622725375063063865962, 10.36238114082703544157302277863

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