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Properties

Label 1-953-953.16-r0-0-0
Degree $1$
Conductor $953$
Sign $0.663 - 0.748i$
Analytic cond. $4.42571$
Root an. cond. $4.42571$
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  + (−0.273 − 0.961i)2-s + (−0.982 − 0.183i)3-s + (−0.850 + 0.526i)4-s + (0.739 − 0.673i)5-s + (0.0922 + 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.739 + 0.673i)8-s + (0.932 + 0.361i)9-s + (−0.850 − 0.526i)10-s + (−0.602 − 0.798i)11-s + (0.932 − 0.361i)12-s + (0.932 + 0.361i)13-s + (−0.602 + 0.798i)14-s + (−0.850 + 0.526i)15-s + (0.445 − 0.895i)16-s + (0.0922 + 0.995i)17-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.982 − 0.183i)3-s + (−0.850 + 0.526i)4-s + (0.739 − 0.673i)5-s + (0.0922 + 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.739 + 0.673i)8-s + (0.932 + 0.361i)9-s + (−0.850 − 0.526i)10-s + (−0.602 − 0.798i)11-s + (0.932 − 0.361i)12-s + (0.932 + 0.361i)13-s + (−0.602 + 0.798i)14-s + (−0.850 + 0.526i)15-s + (0.445 − 0.895i)16-s + (0.0922 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(953\)
Sign: $0.663 - 0.748i$
Analytic conductor: \(4.42571\)
Root analytic conductor: \(4.42571\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{953} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 953,\ (0:\ ),\ 0.663 - 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7777300809 - 0.3500071988i\)
\(L(\frac12)\) \(\approx\) \(0.7777300809 - 0.3500071988i\)
\(L(1)\) \(\approx\) \(0.6308175522 - 0.3552650250i\)
\(L(1)\) \(\approx\) \(0.6308175522 - 0.3552650250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad953 \( 1 \)
good2 \( 1 + (-0.273 - 0.961i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (0.739 - 0.673i)T \)
7 \( 1 + (-0.602 - 0.798i)T \)
11 \( 1 + (-0.602 - 0.798i)T \)
13 \( 1 + (0.932 + 0.361i)T \)
17 \( 1 + (0.0922 + 0.995i)T \)
19 \( 1 + (0.932 + 0.361i)T \)
23 \( 1 + (0.445 + 0.895i)T \)
29 \( 1 + (0.0922 + 0.995i)T \)
31 \( 1 + (0.932 + 0.361i)T \)
37 \( 1 + (-0.273 + 0.961i)T \)
41 \( 1 + (-0.850 + 0.526i)T \)
43 \( 1 + (-0.273 + 0.961i)T \)
47 \( 1 + (0.0922 - 0.995i)T \)
53 \( 1 + (0.0922 + 0.995i)T \)
59 \( 1 + (0.0922 + 0.995i)T \)
61 \( 1 + (-0.602 + 0.798i)T \)
67 \( 1 + (0.932 + 0.361i)T \)
71 \( 1 + (0.932 + 0.361i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (0.932 - 0.361i)T \)
83 \( 1 + (-0.602 + 0.798i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (-0.602 - 0.798i)T \)
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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

−22.35724018406891677510743641692, −21.33377760152520149130301569988, −20.500336023512463542650084180511, −18.85849587748777102135647486976, −18.5633122889761117568359401579, −17.84012982551765399209554942553, −17.319379669125451228417511464418, −16.23028953179796740347052539510, −15.62897224322707677785853186950, −15.167481698739687656672868256675, −13.947096511459746318957244786137, −13.22738591786969846018914954501, −12.39782154401407484796681773774, −11.26939525837749005835761599066, −10.31369132706691665527126126546, −9.75239354315598006286472244474, −9.04594765943152823268846570411, −7.72559869903285147068719209061, −6.79479675087313180404247137256, −6.273503338135113290172570813898, −5.41976592732032030396250268334, −4.87898812980451654889487509976, −3.4323286165294207933917865222, −2.146061087784292074717968853964, −0.59384398248741658513053397703, 1.10415763480015147318704138469, 1.39669612065175344716387661516, 3.04761866755792724845898555549, 3.96148913323212689882520026609, 5.0118427326681631880912744806, 5.76847884129134487730335996583, 6.71937844303124528801270968377, 7.942730170941350708220020302634, 8.79626089935066900426895003742, 9.88555914811497543860988953275, 10.353192537158115296664337254947, 11.13781490624483191192534851429, 11.987651376266861698049267821602, 12.88720540234212498601358134027, 13.44359507614630188639727612917, 13.8787377915571636157229944058, 15.73079366918436820313463822499, 16.65158032967822892051684423600, 16.86342387494713487335674178638, 17.87913802787881681902122138378, 18.45562346051156159270177696063, 19.296977249604737256567492209067, 20.150743925473867575687704520435, 21.05443765299304908557243551324, 21.54375330082749518035455878921

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