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Properties

Label 2-804-201.140-c0-0-0
Degree $2$
Conductor $804$
Sign $0.283 + 0.959i$
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  + (0.415 − 0.909i)3-s + (−0.279 + 0.0538i)7-s + (−0.654 − 0.755i)9-s + (0.0934 − 1.96i)13-s + (0.815 + 0.157i)19-s + (−0.0671 + 0.276i)21-s + (0.841 + 0.540i)25-s + (−0.959 + 0.281i)27-s + (0.0688 + 1.44i)31-s + (−0.841 − 1.45i)37-s + (−1.74 − 0.899i)39-s + (0.252 + 1.75i)43-s + (−0.853 + 0.341i)49-s + (0.481 − 0.676i)57-s + (0.839 + 0.800i)61-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)3-s + (−0.279 + 0.0538i)7-s + (−0.654 − 0.755i)9-s + (0.0934 − 1.96i)13-s + (0.815 + 0.157i)19-s + (−0.0671 + 0.276i)21-s + (0.841 + 0.540i)25-s + (−0.959 + 0.281i)27-s + (0.0688 + 1.44i)31-s + (−0.841 − 1.45i)37-s + (−1.74 − 0.899i)39-s + (0.252 + 1.75i)43-s + (−0.853 + 0.341i)49-s + (0.481 − 0.676i)57-s + (0.839 + 0.800i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.283 + 0.959i$
Analytic conductor: \(0.401248\)
Root analytic conductor: \(0.633441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :0),\ 0.283 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056673561\)
\(L(\frac12)\) \(\approx\) \(1.056673561\)
\(L(1)\) 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 \( 1 + (-0.415 + 0.909i)T \)
67 \( 1 + (0.888 - 0.458i)T \)
good5 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.279 - 0.0538i)T + (0.928 - 0.371i)T^{2} \)
11 \( 1 + (-0.0475 + 0.998i)T^{2} \)
13 \( 1 + (-0.0934 + 1.96i)T + (-0.995 - 0.0950i)T^{2} \)
17 \( 1 + (-0.235 + 0.971i)T^{2} \)
19 \( 1 + (-0.815 - 0.157i)T + (0.928 + 0.371i)T^{2} \)
23 \( 1 + (-0.981 + 0.189i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.0688 - 1.44i)T + (-0.995 + 0.0950i)T^{2} \)
37 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.723 - 0.690i)T^{2} \)
43 \( 1 + (-0.252 - 1.75i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (0.327 - 0.945i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (-0.839 - 0.800i)T + (0.0475 + 0.998i)T^{2} \)
71 \( 1 + (-0.235 - 0.971i)T^{2} \)
73 \( 1 + (-0.341 - 0.325i)T + (0.0475 + 0.998i)T^{2} \)
79 \( 1 + (0.0845 - 0.0436i)T + (0.580 - 0.814i)T^{2} \)
83 \( 1 + (0.888 - 0.458i)T^{2} \)
89 \( 1 + (0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.30733719791212345466996458790, −9.315694043841253540703728245241, −8.469773229158390812400570025500, −7.71416923387730355533742685081, −6.99628118284167383951821097866, −5.93652516866010408739127818790, −5.16917919008808561583328792957, −3.43902257887925033720577609341, −2.80492969136771456808972085494, −1.17102157284832150061581475688, 2.07974918823915172232887437375, 3.33859879913978835640863131191, 4.27445176792782528164140271009, 5.07731016315054229934996946638, 6.30797113724789624603907164439, 7.18473926779504663992110813461, 8.332263946025191027422456882930, 9.095752090204508271756363558350, 9.684877659331258601085254360029, 10.48946349329573243342531450046

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