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Properties

Label 1-804-804.23-r0-0-0
Degree $1$
Conductor $804$
Sign $0.989 - 0.141i$
Analytic cond. $3.73376$
Root an. cond. $3.73376$
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)5-s + (−0.0475 + 0.998i)7-s + (0.981 + 0.189i)11-s + (0.928 + 0.371i)13-s + (−0.580 − 0.814i)17-s + (−0.0475 − 0.998i)19-s + (0.723 − 0.690i)23-s + (−0.142 − 0.989i)25-s + (0.5 + 0.866i)29-s + (−0.928 + 0.371i)31-s + (0.723 + 0.690i)35-s + (−0.5 + 0.866i)37-s + (0.995 − 0.0950i)41-s + (−0.415 + 0.909i)43-s + (0.235 + 0.971i)47-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)5-s + (−0.0475 + 0.998i)7-s + (0.981 + 0.189i)11-s + (0.928 + 0.371i)13-s + (−0.580 − 0.814i)17-s + (−0.0475 − 0.998i)19-s + (0.723 − 0.690i)23-s + (−0.142 − 0.989i)25-s + (0.5 + 0.866i)29-s + (−0.928 + 0.371i)31-s + (0.723 + 0.690i)35-s + (−0.5 + 0.866i)37-s + (0.995 − 0.0950i)41-s + (−0.415 + 0.909i)43-s + (0.235 + 0.971i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(3.73376\)
Root analytic conductor: \(3.73376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 804,\ (0:\ ),\ 0.989 - 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.806614883 - 0.1284550877i\)
\(L(\frac12)\) \(\approx\) \(1.806614883 - 0.1284550877i\)
\(L(1)\) \(\approx\) \(1.301210940 - 0.05200355248i\)
\(L(1)\) \(\approx\) \(1.301210940 - 0.05200355248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
67 \( 1 \)
good5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.0475 + 0.998i)T \)
11 \( 1 + (0.981 + 0.189i)T \)
13 \( 1 + (0.928 + 0.371i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.235 + 0.971i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (0.981 - 0.189i)T \)
79 \( 1 + (0.786 - 0.618i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.34118772756379548454136623213, −21.466301895968181873468893628197, −20.74820546123183272453057314160, −19.82428574847898018090958083243, −19.09870110685011868562596857856, −18.222583427245977898185823470394, −17.3055858718773898662676592601, −16.93629418741630555839237151711, −15.77664161185615994720461549784, −14.84083314057473760349613636234, −14.07187060423365281215563260073, −13.49138585778315490495580630654, −12.61913046806790301148625435802, −11.28573203993803560458066367831, −10.79264608468882729191579595369, −9.97109614547176930545038252421, −9.082081915369725925486762403379, −8.0196533062810383867431576508, −7.00454467923052928889038466604, −6.315330459642823802845694994215, −5.51474715980122320371553635055, −3.90955499867455301934435187659, −3.593168233138516963737050294472, −2.09616259504457319230403369658, −1.12502135193626299799691265521, 1.08156581489467879282492188757, 2.07193229238483226647381384997, 3.125378749654730929577529545451, 4.51207082791464729027162729891, 5.15446014181220927682513195796, 6.26526320375052489115181998211, 6.83465721153085298743861402027, 8.392358795014022392587840055587, 9.11168859817208445189042850454, 9.37093694267531784691035318304, 10.81906486280679969440871489748, 11.62415333410760408119994535218, 12.48559325414921017829722313484, 13.20288398307059262038734221743, 14.07430279097665594649973282824, 14.91423267268776913635543199211, 15.97205912023073744055332971151, 16.447574592600161265103268342477, 17.58166792804893684609657679140, 18.05441975748938566181479245179, 19.02581812748529622192238436278, 19.92865136060107883002055861941, 20.680798705045287605637044780603, 21.47631415452875008678648874469, 22.11041015775104196760368634895

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