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Properties

Label 2-804-201.188-c0-0-0
Degree $2$
Conductor $804$
Sign $-0.817 + 0.575i$
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  + (−0.654 + 0.755i)3-s + (−1.78 − 0.713i)7-s + (−0.142 − 0.989i)9-s + (−1.84 + 0.176i)13-s + (−1.21 + 0.486i)19-s + (1.70 − 0.879i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−0.0947 − 0.00904i)31-s + (−0.415 − 0.719i)37-s + (1.07 − 1.51i)39-s + (−1.11 − 0.326i)43-s + (1.94 + 1.85i)49-s + (0.428 − 1.23i)57-s + (−0.0311 + 0.653i)61-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)3-s + (−1.78 − 0.713i)7-s + (−0.142 − 0.989i)9-s + (−1.84 + 0.176i)13-s + (−1.21 + 0.486i)19-s + (1.70 − 0.879i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−0.0947 − 0.00904i)31-s + (−0.415 − 0.719i)37-s + (1.07 − 1.51i)39-s + (−1.11 − 0.326i)43-s + (1.94 + 1.85i)49-s + (0.428 − 1.23i)57-s + (−0.0311 + 0.653i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(0.401248\)
Root analytic conductor: \(0.633441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :0),\ -0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07085363766\)
\(L(\frac12)\) \(\approx\) \(0.07085363766\)
\(L(1)\) 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 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
good5 \( 1 + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (1.78 + 0.713i)T + (0.723 + 0.690i)T^{2} \)
11 \( 1 + (0.995 - 0.0950i)T^{2} \)
13 \( 1 + (1.84 - 0.176i)T + (0.981 - 0.189i)T^{2} \)
17 \( 1 + (0.888 - 0.458i)T^{2} \)
19 \( 1 + (1.21 - 0.486i)T + (0.723 - 0.690i)T^{2} \)
23 \( 1 + (-0.928 - 0.371i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.0947 + 0.00904i)T + (0.981 + 0.189i)T^{2} \)
37 \( 1 + (0.415 + 0.719i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.0475 + 0.998i)T^{2} \)
43 \( 1 + (1.11 + 0.326i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.786 - 0.618i)T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (0.0311 - 0.653i)T + (-0.995 - 0.0950i)T^{2} \)
71 \( 1 + (0.888 + 0.458i)T^{2} \)
73 \( 1 + (0.0845 - 1.77i)T + (-0.995 - 0.0950i)T^{2} \)
79 \( 1 + (1.15 + 1.62i)T + (-0.327 + 0.945i)T^{2} \)
83 \( 1 + (-0.580 - 0.814i)T^{2} \)
89 \( 1 + (0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.10705407753663906938624361824, −9.642259803914182893985597164102, −8.678696166762429505098501282922, −7.19807796642669736926299128956, −6.65307353239513138322182539749, −5.75628197610029297159132000856, −4.59556424378235864888375588361, −3.80943809978851047755496325403, −2.68429372184684496228467490814, −0.07145287117382331491844641735, 2.23035480279391141555863332827, 3.13038467398843644213422975509, 4.79031317376222176044262898515, 5.64975353313503496422065975309, 6.65121482210047489679564943262, 6.99892634373789375070053269136, 8.182708334998171153822149094449, 9.266260193570025961979737140673, 9.923268018952180336787588168805, 10.78333257166456479703168808552

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