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Properties

Label 1-828-828.355-r0-0-0
Degree $1$
Conductor $828$
Sign $-0.0384 - 0.999i$
Analytic cond. $3.84521$
Root an. cond. $3.84521$
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  + (0.327 − 0.945i)5-s + (−0.888 + 0.458i)7-s + (0.928 + 0.371i)11-s + (−0.888 − 0.458i)13-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.786 − 0.618i)25-s + (0.235 − 0.971i)29-s + (0.995 − 0.0950i)31-s + (0.142 + 0.989i)35-s + (0.654 − 0.755i)37-s + (−0.327 + 0.945i)41-s + (−0.995 − 0.0950i)43-s + (0.5 − 0.866i)47-s + (0.580 − 0.814i)49-s + ⋯
L(s)  = 1  + (0.327 − 0.945i)5-s + (−0.888 + 0.458i)7-s + (0.928 + 0.371i)11-s + (−0.888 − 0.458i)13-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.786 − 0.618i)25-s + (0.235 − 0.971i)29-s + (0.995 − 0.0950i)31-s + (0.142 + 0.989i)35-s + (0.654 − 0.755i)37-s + (−0.327 + 0.945i)41-s + (−0.995 − 0.0950i)43-s + (0.5 − 0.866i)47-s + (0.580 − 0.814i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0384 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0384 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.0384 - 0.999i$
Analytic conductor: \(3.84521\)
Root analytic conductor: \(3.84521\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (0:\ ),\ -0.0384 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7988650461 - 0.8302012184i\)
\(L(\frac12)\) \(\approx\) \(0.7988650461 - 0.8302012184i\)
\(L(1)\) \(\approx\) \(0.9557665250 - 0.2578565994i\)
\(L(1)\) \(\approx\) \(0.9557665250 - 0.2578565994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.327 - 0.945i)T \)
7 \( 1 + (-0.888 + 0.458i)T \)
11 \( 1 + (0.928 + 0.371i)T \)
13 \( 1 + (-0.888 - 0.458i)T \)
17 \( 1 + (0.959 - 0.281i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (0.235 - 0.971i)T \)
31 \( 1 + (0.995 - 0.0950i)T \)
37 \( 1 + (0.654 - 0.755i)T \)
41 \( 1 + (-0.327 + 0.945i)T \)
43 \( 1 + (-0.995 - 0.0950i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (0.888 + 0.458i)T \)
61 \( 1 + (-0.580 - 0.814i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (0.0475 + 0.998i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.981 + 0.189i)T \)
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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

−22.12231293727002737752890605511, −21.929042778258605033626444232952, −20.86039019937621355417764978755, −19.73821647694851111409938465886, −19.1435064998244757527830882029, −18.660847306871424378888114073330, −17.30383070190703923524341016512, −16.96214196398559125964569884777, −16.02320221022398244412048814293, −14.88232040670362441088261340551, −14.34562986020128254535125384193, −13.61316420168370884096825640029, −12.56112598891613732771365431204, −11.79998052761451765457738465639, −10.7336271271520471477875691871, −10.03549644832291171422437131789, −9.39222662350290244842414099496, −8.22074363666185853476707431104, −7.04295336213618159050107960242, −6.58308276718109000384548963094, −5.7298749591816289610481056010, −4.329157125052927869680789698584, −3.43548415843884282218959866139, −2.61970073035850874043887829446, −1.32309255047923907009065845088, 0.542968232403772838704713758529, 1.89200629485448309600621695268, 2.911466489805261828243139691352, 4.118614915546268806611620398475, 5.00449630835692861385625442875, 5.956227484880604831402872097310, 6.72414846893447422034949844073, 7.91773599973329661386115804023, 8.7900989514995878934472032724, 9.72304591884803967091286027406, 10.00440405052462605011090425605, 11.6076971558392459927137521225, 12.29952972141066411719068514126, 12.85925089057637000188177481308, 13.75123085887252611068141286443, 14.79460273404435100187530012263, 15.50487911957534237288651633178, 16.56695667809652491119693458653, 16.994004314615967779901021477390, 17.80934835627188136557507672129, 18.98043228900671622442711057882, 19.61614443572064513376869195437, 20.252794360717215289723429981891, 21.29194921005120080063365751423, 21.85732388563650779640080647012

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