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Properties

Label 2-804-201.167-c0-0-0
Degree $2$
Conductor $804$
Sign $0.970 - 0.241i$
Analytic cond. $0.401248$
Root an. cond. $0.633441$
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  + (−0.959 + 0.281i)3-s + (−0.738 − 0.380i)7-s + (0.841 − 0.540i)9-s + (1.39 + 1.09i)13-s + (1.70 − 0.879i)19-s + (0.815 + 0.157i)21-s + (−0.142 + 0.989i)25-s + (−0.654 + 0.755i)27-s + (0.514 − 0.404i)31-s + (0.142 − 0.246i)37-s + (−1.65 − 0.660i)39-s + (0.771 + 1.68i)43-s + (−0.179 − 0.252i)49-s + (−1.38 + 1.32i)57-s + (−0.473 − 1.36i)61-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)3-s + (−0.738 − 0.380i)7-s + (0.841 − 0.540i)9-s + (1.39 + 1.09i)13-s + (1.70 − 0.879i)19-s + (0.815 + 0.157i)21-s + (−0.142 + 0.989i)25-s + (−0.654 + 0.755i)27-s + (0.514 − 0.404i)31-s + (0.142 − 0.246i)37-s + (−1.65 − 0.660i)39-s + (0.771 + 1.68i)43-s + (−0.179 − 0.252i)49-s + (−1.38 + 1.32i)57-s + (−0.473 − 1.36i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.970 - 0.241i$
Analytic conductor: \(0.401248\)
Root analytic conductor: \(0.633441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :0),\ 0.970 - 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7344953298\)
\(L(\frac12)\) \(\approx\) \(0.7344953298\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.928 + 0.371i)T \)
good5 \( 1 + (0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.738 + 0.380i)T + (0.580 + 0.814i)T^{2} \)
11 \( 1 + (0.786 + 0.618i)T^{2} \)
13 \( 1 + (-1.39 - 1.09i)T + (0.235 + 0.971i)T^{2} \)
17 \( 1 + (-0.981 - 0.189i)T^{2} \)
19 \( 1 + (-1.70 + 0.879i)T + (0.580 - 0.814i)T^{2} \)
23 \( 1 + (0.888 + 0.458i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.514 + 0.404i)T + (0.235 - 0.971i)T^{2} \)
37 \( 1 + (-0.142 + 0.246i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.327 + 0.945i)T^{2} \)
43 \( 1 + (-0.771 - 1.68i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (-0.0475 - 0.998i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.473 + 1.36i)T + (-0.786 + 0.618i)T^{2} \)
71 \( 1 + (-0.981 + 0.189i)T^{2} \)
73 \( 1 + (0.642 + 1.85i)T + (-0.786 + 0.618i)T^{2} \)
79 \( 1 + (1.45 - 0.584i)T + (0.723 - 0.690i)T^{2} \)
83 \( 1 + (-0.928 + 0.371i)T^{2} \)
89 \( 1 + (-0.841 - 0.540i)T^{2} \)
97 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.63973914383996246027699635146, −9.488789377509669544764129530182, −9.281655756778430889507524589648, −7.76099245120920640677501065268, −6.79884385036490309845495916797, −6.21754790524756174130446890965, −5.21249212840163917536763526752, −4.16308996679730464036916232613, −3.26275628916314301062430649135, −1.23885615657346815214178316080, 1.15784804906453481738327318274, 2.94410056890219452685796755460, 4.05624789803210923506716982393, 5.52600140140318359208756051986, 5.84729144165175801629847642912, 6.83440592219767877562539237412, 7.79796325352033541077172172829, 8.677581041598944788225516357342, 9.891189446767500237350418772911, 10.37020188377944756674399234585

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