↑

Properties

Label 1-463-463.8-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

Related objects

Downloads

Learn more

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: $\chi_{463} (8, \cdot )$
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 \( 1 + (-0.818 + 0.574i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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