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Properties

Label 1-19e2-361.139-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: $\chi_{361} (139, \cdot )$
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

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

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