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Properties

Label 1-19e2-361.187-r0-0-0
Degree $1$
Conductor $361$
Sign $0.613 - 0.790i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (0.967 − 0.254i)2-s + (−0.367 + 0.929i)3-s + (0.870 − 0.492i)4-s + (−0.999 − 0.0183i)5-s + (−0.119 + 0.992i)6-s + (0.635 − 0.771i)7-s + (0.716 − 0.697i)8-s + (−0.729 − 0.684i)9-s + (−0.971 + 0.236i)10-s + (0.451 − 0.892i)11-s + (0.137 + 0.990i)12-s + (−0.979 − 0.200i)13-s + (0.418 − 0.908i)14-s + (0.384 − 0.922i)15-s + (0.515 − 0.856i)16-s + (−0.00918 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.967 − 0.254i)2-s + (−0.367 + 0.929i)3-s + (0.870 − 0.492i)4-s + (−0.999 − 0.0183i)5-s + (−0.119 + 0.992i)6-s + (0.635 − 0.771i)7-s + (0.716 − 0.697i)8-s + (−0.729 − 0.684i)9-s + (−0.971 + 0.236i)10-s + (0.451 − 0.892i)11-s + (0.137 + 0.990i)12-s + (−0.979 − 0.200i)13-s + (0.418 − 0.908i)14-s + (0.384 − 0.922i)15-s + (0.515 − 0.856i)16-s + (−0.00918 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.613 - 0.790i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.613 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.571963673 - 0.7698991639i\)
\(L(\frac12)\) \(\approx\) \(1.571963673 - 0.7698991639i\)
\(L(1)\) \(\approx\) \(1.448133734 - 0.2545101431i\)
\(L(1)\) \(\approx\) \(1.448133734 - 0.2545101431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.967 - 0.254i)T \)
3 \( 1 + (-0.367 + 0.929i)T \)
5 \( 1 + (-0.999 - 0.0183i)T \)
7 \( 1 + (0.635 - 0.771i)T \)
11 \( 1 + (0.451 - 0.892i)T \)
13 \( 1 + (-0.979 - 0.200i)T \)
17 \( 1 + (-0.00918 - 0.999i)T \)
23 \( 1 + (0.989 + 0.146i)T \)
29 \( 1 + (-0.333 + 0.942i)T \)
31 \( 1 + (-0.821 - 0.569i)T \)
37 \( 1 + (0.546 + 0.837i)T \)
41 \( 1 + (-0.0459 + 0.998i)T \)
43 \( 1 + (0.690 - 0.723i)T \)
47 \( 1 + (0.919 - 0.393i)T \)
53 \( 1 + (-0.800 - 0.599i)T \)
59 \( 1 + (-0.842 - 0.539i)T \)
61 \( 1 + (0.100 + 0.994i)T \)
67 \( 1 + (0.997 - 0.0734i)T \)
71 \( 1 + (0.100 - 0.994i)T \)
73 \( 1 + (-0.861 + 0.507i)T \)
79 \( 1 + (0.280 + 0.959i)T \)
83 \( 1 + (-0.754 + 0.656i)T \)
89 \( 1 + (-0.861 - 0.507i)T \)
97 \( 1 + (0.997 + 0.0734i)T \)
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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

−24.64302376196997338249729031889, −23.922799739075352930160216896846, −23.212227231268322636837937648664, −22.443054200438955725552890039066, −21.649578613082201904509562780605, −20.4262400182739659786356803567, −19.570431033964390677020893836, −18.85696003402914071450201340266, −17.51271098119103629809986663823, −16.95125835989398704996196393632, −15.67524387917289610691490655568, −14.826261825035635518939928420595, −14.30188549518503838952614556078, −12.66032703102443233429266260126, −12.47940519366341364454401244017, −11.57283175545402043389036972249, −10.85362800283906488248854115811, −8.87315281970062389913062347737, −7.71370280987450089192675340172, −7.251561762342419491429878057759, −6.11832013576904875446096046172, −5.05549747625023472173099641923, −4.193209322114029545959526624467, −2.68502435049830361590914008912, −1.7058492749811568648408851132, 0.85489061155564650658176020385, 2.93659632034575040245615676528, 3.77423763737241173175471319417, 4.67285015075349120721322790624, 5.33339207482591399359446636681, 6.79024737158691748488328118208, 7.69748865264204002114689402275, 9.13369894761451705580533811697, 10.36343710886383287744794644710, 11.293725010681514720759803037455, 11.55269902125318067870502326372, 12.74816218929999600976923145968, 14.07339308540156408774001928949, 14.712900389150800189651690784988, 15.501775760841418374737477679657, 16.51187778078831479602448989958, 16.97843976329223313054766061226, 18.6144756974534239741877153223, 19.82788910356613530854071016255, 20.27322134218828577301612358003, 21.16462832561374563923365207097, 22.13147063992669439647812664238, 22.681233026265306800079631337165, 23.670071499790051707891069876223, 24.12267729106894065267109579332

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