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Properties

Label 2-804-201.71-c0-0-0
Degree $2$
Conductor $804$
Sign $0.578 - 0.815i$
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  + (0.841 + 0.540i)3-s + (−0.759 + 1.06i)7-s + (0.415 + 0.909i)9-s + (0.273 − 1.12i)13-s + (0.975 + 1.37i)19-s + (−1.21 + 0.486i)21-s + (−0.959 + 0.281i)25-s + (−0.142 + 0.989i)27-s + (−0.370 − 1.52i)31-s + (0.959 − 1.66i)37-s + (0.839 − 0.800i)39-s + (−0.947 − 1.09i)43-s + (−0.233 − 0.676i)49-s + (0.0800 + 1.68i)57-s + (−0.0748 − 0.0588i)61-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)3-s + (−0.759 + 1.06i)7-s + (0.415 + 0.909i)9-s + (0.273 − 1.12i)13-s + (0.975 + 1.37i)19-s + (−1.21 + 0.486i)21-s + (−0.959 + 0.281i)25-s + (−0.142 + 0.989i)27-s + (−0.370 − 1.52i)31-s + (0.959 − 1.66i)37-s + (0.839 − 0.800i)39-s + (−0.947 − 1.09i)43-s + (−0.233 − 0.676i)49-s + (0.0800 + 1.68i)57-s + (−0.0748 − 0.0588i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(0.401248\)
Root analytic conductor: \(0.633441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :0),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196401238\)
\(L(\frac12)\) \(\approx\) \(1.196401238\)
\(L(1)\) 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 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
good5 \( 1 + (0.959 - 0.281i)T^{2} \)
7 \( 1 + (0.759 - 1.06i)T + (-0.327 - 0.945i)T^{2} \)
11 \( 1 + (-0.235 + 0.971i)T^{2} \)
13 \( 1 + (-0.273 + 1.12i)T + (-0.888 - 0.458i)T^{2} \)
17 \( 1 + (-0.928 + 0.371i)T^{2} \)
19 \( 1 + (-0.975 - 1.37i)T + (-0.327 + 0.945i)T^{2} \)
23 \( 1 + (-0.580 + 0.814i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.370 + 1.52i)T + (-0.888 + 0.458i)T^{2} \)
37 \( 1 + (-0.959 + 1.66i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.786 + 0.618i)T^{2} \)
43 \( 1 + (0.947 + 1.09i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (0.995 + 0.0950i)T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (0.0748 + 0.0588i)T + (0.235 + 0.971i)T^{2} \)
71 \( 1 + (-0.928 - 0.371i)T^{2} \)
73 \( 1 + (1.45 + 1.14i)T + (0.235 + 0.971i)T^{2} \)
79 \( 1 + (-0.341 - 0.325i)T + (0.0475 + 0.998i)T^{2} \)
83 \( 1 + (-0.723 - 0.690i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.28397942254465998181362463682, −9.684449476208998409220936597812, −9.035593404315920968243398909560, −8.082642071487100074151863534160, −7.48064896816154785501577584614, −5.90791088177237949383071936815, −5.47906441468394844154384613002, −3.94057212250338416456528484112, −3.18926149762951424855899533183, −2.12926044543944319266448318066, 1.33101957955815381094738187114, 2.84911245774210637329482784935, 3.73026952960960648875960222738, 4.74068782029250800365891067310, 6.40854186032612909547852537029, 6.89049200975144295688015873948, 7.67439987817380811671238166820, 8.675289909121623809314957749430, 9.505294404562112815012686217844, 10.04759123349513357881469894389

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