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Properties

Label 1-463-463.58-r0-0-0
Degree $1$
Conductor $463$
Sign $0.340 + 0.940i$
Analytic cond. $2.15016$
Root an. cond. $2.15016$
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  + (−0.999 + 0.0407i)2-s + (0.182 − 0.983i)3-s + (0.996 − 0.0815i)4-s + (−0.818 + 0.574i)5-s + (−0.142 + 0.989i)6-s + (0.182 + 0.983i)7-s + (−0.992 + 0.122i)8-s + (−0.933 − 0.359i)9-s + (0.794 − 0.607i)10-s + (−0.999 − 0.0407i)11-s + (0.101 − 0.994i)12-s + (0.623 − 0.781i)13-s + (−0.222 − 0.974i)14-s + (0.415 + 0.909i)15-s + (0.986 − 0.162i)16-s + (−0.591 − 0.806i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0407i)2-s + (0.182 − 0.983i)3-s + (0.996 − 0.0815i)4-s + (−0.818 + 0.574i)5-s + (−0.142 + 0.989i)6-s + (0.182 + 0.983i)7-s + (−0.992 + 0.122i)8-s + (−0.933 − 0.359i)9-s + (0.794 − 0.607i)10-s + (−0.999 − 0.0407i)11-s + (0.101 − 0.994i)12-s + (0.623 − 0.781i)13-s + (−0.222 − 0.974i)14-s + (0.415 + 0.909i)15-s + (0.986 − 0.162i)16-s + (−0.591 − 0.806i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(463\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(2.15016\)
Root analytic conductor: \(2.15016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{463} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 463,\ (0:\ ),\ 0.340 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3844685305 + 0.2697602387i\)
\(L(\frac12)\) \(\approx\) \(0.3844685305 + 0.2697602387i\)
\(L(1)\) \(\approx\) \(0.5599281296 + 0.004860958561i\)
\(L(1)\) \(\approx\) \(0.5599281296 + 0.004860958561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad463 \( 1 \)
good2 \( 1 + (-0.999 + 0.0407i)T \)
3 \( 1 + (0.182 - 0.983i)T \)
5 \( 1 + (-0.818 + 0.574i)T \)
7 \( 1 + (0.182 + 0.983i)T \)
11 \( 1 + (-0.999 - 0.0407i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.591 - 0.806i)T \)
19 \( 1 + (0.970 - 0.242i)T \)
23 \( 1 + (-0.768 + 0.639i)T \)
29 \( 1 + (0.986 + 0.162i)T \)
31 \( 1 + (-0.377 + 0.925i)T \)
37 \( 1 + (0.339 + 0.940i)T \)
41 \( 1 + (-0.0611 + 0.998i)T \)
43 \( 1 + (-0.768 + 0.639i)T \)
47 \( 1 + (-0.452 + 0.891i)T \)
53 \( 1 + (0.557 + 0.830i)T \)
59 \( 1 + (0.882 + 0.470i)T \)
61 \( 1 + (-0.452 + 0.891i)T \)
67 \( 1 + (0.947 + 0.320i)T \)
71 \( 1 + (0.882 - 0.470i)T \)
73 \( 1 + (-0.979 - 0.202i)T \)
79 \( 1 + (0.101 + 0.994i)T \)
83 \( 1 + (-0.591 + 0.806i)T \)
89 \( 1 + (-0.591 - 0.806i)T \)
97 \( 1 + (-0.818 - 0.574i)T \)
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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

−23.80693197493511210175475849102, −23.07911453726392013416455739608, −21.70238518436747428737370616000, −20.80564508985430750321288756297, −20.31074848170903331894310949115, −19.68880376887605823918529751643, −18.66417025017126012504621751576, −17.58599819326099782828252291501, −16.65212327504514125415541064968, −16.1042717354886489317793312871, −15.521574497283358167191866296854, −14.434171846605718199905661716617, −13.26812398775919693060271940511, −11.93893236637596472974743140672, −11.09729190963942891293958145226, −10.442500486730711891512108993122, −9.564824716918644960584043936272, −8.4404471121741152524890634364, −8.0384226871054022692793049505, −6.8858572517730951687274310188, −5.475492802022341959518057674334, −4.228783013013092432531897048127, −3.53133714419650005190223256446, −2.023274697835913439451680584442, −0.38407285503158431410685236258, 1.2051409002906353574609289049, 2.69700536812161694190605204655, 3.03066095771184803439836176174, 5.28791977281286341517030173973, 6.304253774642184516913389490075, 7.25271615156950103991433710609, 8.06334552968116679002697683408, 8.540384016031056777240050138368, 9.800185604509806501667635423414, 11.03419305122517830289353350861, 11.64187599650566769643936048606, 12.40879589019114115945146410307, 13.57204229133015614601033030022, 14.77699767910548595814694379011, 15.63568844965546134102507127202, 16.10474159198140854762044471581, 17.8554845879982310678081054358, 18.13840152361002125263684434773, 18.624774751817982546400621537190, 19.74646458479599340922265060627, 20.145850471613560179348484939739, 21.320257938139569457526366458660, 22.53189538905391049602762016715, 23.50710092775121019779386166295, 24.185543727858590717820008040282

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