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Properties

Label 1-804-804.467-r0-0-0
Degree $1$
Conductor $804$
Sign $-0.869 - 0.494i$
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  + (0.142 + 0.989i)5-s + (−0.580 + 0.814i)7-s + (−0.786 + 0.618i)11-s + (0.235 − 0.971i)13-s + (−0.981 + 0.189i)17-s + (−0.580 − 0.814i)19-s + (−0.888 + 0.458i)23-s + (−0.959 + 0.281i)25-s + (0.5 + 0.866i)29-s + (−0.235 − 0.971i)31-s + (−0.888 − 0.458i)35-s + (−0.5 + 0.866i)37-s + (0.327 − 0.945i)41-s + (0.654 + 0.755i)43-s + (0.0475 − 0.998i)47-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)5-s + (−0.580 + 0.814i)7-s + (−0.786 + 0.618i)11-s + (0.235 − 0.971i)13-s + (−0.981 + 0.189i)17-s + (−0.580 − 0.814i)19-s + (−0.888 + 0.458i)23-s + (−0.959 + 0.281i)25-s + (0.5 + 0.866i)29-s + (−0.235 − 0.971i)31-s + (−0.888 − 0.458i)35-s + (−0.5 + 0.866i)37-s + (0.327 − 0.945i)41-s + (0.654 + 0.755i)43-s + (0.0475 − 0.998i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.869 - 0.494i$
Analytic conductor: \(3.73376\)
Root analytic conductor: \(3.73376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 804,\ (0:\ ),\ -0.869 - 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04743271530 + 0.1791996037i\)
\(L(\frac12)\) \(\approx\) \(-0.04743271530 + 0.1791996037i\)
\(L(1)\) \(\approx\) \(0.7055512474 + 0.2102875381i\)
\(L(1)\) \(\approx\) \(0.7055512474 + 0.2102875381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
67 \( 1 \)
good5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.580 + 0.814i)T \)
11 \( 1 + (-0.786 + 0.618i)T \)
13 \( 1 + (0.235 - 0.971i)T \)
17 \( 1 + (-0.981 + 0.189i)T \)
19 \( 1 + (-0.580 - 0.814i)T \)
23 \( 1 + (-0.888 + 0.458i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.235 - 0.971i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.327 - 0.945i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.0475 - 0.998i)T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (0.981 + 0.189i)T \)
73 \( 1 + (-0.786 - 0.618i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (0.928 + 0.371i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
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

−21.52354012382976482332339796591, −21.06661680490797570304056844323, −20.15551936674780909991232957551, −19.52968375365896743791245634251, −18.64619536749582199813399151435, −17.66192383819156634828020450323, −16.80406100641208630841278676409, −16.18004942044071325074204445017, −15.6751229770024445502070087755, −14.132688395339163879792651473857, −13.678069562377038504697193850476, −12.83643721377371473319985836285, −12.11533265198208125293407822774, −10.96581464898096074953774886723, −10.23747081175703172905514403116, −9.248250148312033428403438955, −8.516303623328656225038480305228, −7.61984752151108566081417169100, −6.4855604054441229168675795995, −5.75669375452679880872219383450, −4.47855412375337957250379707865, −3.98780833897493890858326030465, −2.57972415245178060350437512793, −1.417716754200990082907557911742, −0.08016056452710826786883048338, 2.086889015732773747955036865427, 2.696444828810811781988797630384, 3.67443769611095442827656475994, 4.9860818899851853009463759179, 5.94354790124042063387035641106, 6.67281704237233290525140172074, 7.61451064577839591449657118263, 8.58727964507286850493840276071, 9.59274925660220283336225259025, 10.4045453816865409754338196716, 11.061856266172865769945317826320, 12.1468051152679702071782997424, 13.001776083149605066225502279751, 13.64384394507540802403274060348, 14.905769508996398299295154960190, 15.390746051756032839315940769969, 15.93732074174784400848645883567, 17.38768843856249674500970704460, 17.99525740008189944812813112135, 18.568216118779706008664129181086, 19.51705693793233484403699258876, 20.19396921881875738458055801681, 21.314213222621238147566632008512, 22.04945593655196131473134588383, 22.55747251394057853130071213053

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