Properties

Label 2-804-201.35-c0-0-0
Degree $2$
Conductor $804$
Sign $0.865 + 0.500i$
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  + (−0.142 − 0.989i)3-s + (1.21 + 1.16i)7-s + (−0.959 + 0.281i)9-s + (1.42 − 0.273i)13-s + (−0.205 + 0.196i)19-s + (0.975 − 1.37i)21-s + (−0.654 − 0.755i)25-s + (0.415 + 0.909i)27-s + (−1.95 − 0.376i)31-s + (0.654 − 1.13i)37-s + (−0.473 − 1.36i)39-s + (−0.550 − 0.353i)43-s + (0.0871 + 1.82i)49-s + (0.223 + 0.175i)57-s + (1.56 + 0.149i)61-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)3-s + (1.21 + 1.16i)7-s + (−0.959 + 0.281i)9-s + (1.42 − 0.273i)13-s + (−0.205 + 0.196i)19-s + (0.975 − 1.37i)21-s + (−0.654 − 0.755i)25-s + (0.415 + 0.909i)27-s + (−1.95 − 0.376i)31-s + (0.654 − 1.13i)37-s + (−0.473 − 1.36i)39-s + (−0.550 − 0.353i)43-s + (0.0871 + 1.82i)49-s + (0.223 + 0.175i)57-s + (1.56 + 0.149i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(0.401248\)
Root analytic conductor: \(0.633441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :0),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045789959\)
\(L(\frac12)\) \(\approx\) \(1.045789959\)
\(L(1)\) 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 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
good5 \( 1 + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (-1.21 - 1.16i)T + (0.0475 + 0.998i)T^{2} \)
11 \( 1 + (-0.981 + 0.189i)T^{2} \)
13 \( 1 + (-1.42 + 0.273i)T + (0.928 - 0.371i)T^{2} \)
17 \( 1 + (-0.580 + 0.814i)T^{2} \)
19 \( 1 + (0.205 - 0.196i)T + (0.0475 - 0.998i)T^{2} \)
23 \( 1 + (-0.723 - 0.690i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.95 + 0.376i)T + (0.928 + 0.371i)T^{2} \)
37 \( 1 + (-0.654 + 1.13i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.995 + 0.0950i)T^{2} \)
43 \( 1 + (0.550 + 0.353i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.235 + 0.971i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (-1.56 - 0.149i)T + (0.981 + 0.189i)T^{2} \)
71 \( 1 + (-0.580 - 0.814i)T^{2} \)
73 \( 1 + (1.15 + 0.110i)T + (0.981 + 0.189i)T^{2} \)
79 \( 1 + (0.642 - 1.85i)T + (-0.786 - 0.618i)T^{2} \)
83 \( 1 + (0.327 - 0.945i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.76058982120291004441956158311, −9.268263715686660478495297889273, −8.427994619613486396290813862243, −8.055307808844688319425111228598, −6.96749221785038834535100289845, −5.77504781547844487158596933804, −5.52032789238713373489315621676, −3.98044863221021590497162802478, −2.46797091864087138532156815064, −1.54767577562072779371598694652, 1.53755552265911241379666092597, 3.45355542820890897623435586595, 4.16774655310867951341212936337, 5.02411455058047808060180171951, 5.99923676867595261789915870971, 7.16010556964979231748794909792, 8.140401800015385372060743858326, 8.835270956521519846076621609902, 9.801177517499693176604966553885, 10.71951297644070887249509252967

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