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Properties

Label 1-953-953.238-r0-0-0
Degree $1$
Conductor $953$
Sign $-0.622 + 0.782i$
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  + (−0.602 − 0.798i)2-s + (0.0922 + 0.995i)3-s + (−0.273 + 0.961i)4-s + (0.932 + 0.361i)5-s + (0.739 − 0.673i)6-s + (0.445 + 0.895i)7-s + (0.932 − 0.361i)8-s + (−0.982 + 0.183i)9-s + (−0.273 − 0.961i)10-s + (0.445 + 0.895i)11-s + (−0.982 − 0.183i)12-s + (−0.982 + 0.183i)13-s + (0.445 − 0.895i)14-s + (−0.273 + 0.961i)15-s + (−0.850 − 0.526i)16-s + (0.739 − 0.673i)17-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (0.0922 + 0.995i)3-s + (−0.273 + 0.961i)4-s + (0.932 + 0.361i)5-s + (0.739 − 0.673i)6-s + (0.445 + 0.895i)7-s + (0.932 − 0.361i)8-s + (−0.982 + 0.183i)9-s + (−0.273 − 0.961i)10-s + (0.445 + 0.895i)11-s + (−0.982 − 0.183i)12-s + (−0.982 + 0.183i)13-s + (0.445 − 0.895i)14-s + (−0.273 + 0.961i)15-s + (−0.850 − 0.526i)16-s + (0.739 − 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(953\)
Sign: $-0.622 + 0.782i$
Analytic conductor: \(4.42571\)
Root analytic conductor: \(4.42571\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{953} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 953,\ (0:\ ),\ -0.622 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4413703621 + 0.9147623933i\)
\(L(\frac12)\) \(\approx\) \(0.4413703621 + 0.9147623933i\)
\(L(1)\) \(\approx\) \(0.8105306992 + 0.3136154307i\)
\(L(1)\) \(\approx\) \(0.8105306992 + 0.3136154307i\)

Euler product

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

−21.560642132708524636737989521734, −20.40654766672797963470371204250, −19.72191879408544228870678251054, −19.06872873689775378340703810160, −18.142920495549567690995324762544, −17.48970981326543466133581186303, −16.8745208354880491002524520408, −16.507438200771431310620213063598, −14.88203298369475561445364886074, −14.26429374424265302284788888816, −13.797000055545471556719204235410, −12.91521103498039077817042264766, −11.98314899443507437056151315387, −10.63283860255330099856720230444, −10.22957646232161611028769408198, −8.860493317003830526775946937884, −8.50222019990149923308479062583, −7.44846800561246190124692219502, −6.80659126803948013399217365037, −5.89140439624830512542673163978, −5.28233069457413674582422706294, −3.97724254517984400835296152535, −2.29826488438483356555421305955, −1.478458617983642592317102255523, −0.51695583733662540274911934784, 1.75561462745232627612843624841, 2.38448240175927228047875011525, 3.271794586426353229855801637404, 4.50989845909581797574512070255, 5.15458167250947455371828819288, 6.32770487410386248518095114265, 7.56873194073705603121521457190, 8.55701553590593065362697373185, 9.425205666377574073095028176378, 9.8437819758624302096059865277, 10.48786574844408457850977419835, 11.64672939781827515469637814928, 12.05253382903701615946645294088, 13.17515346560489621884031639958, 14.384065675962183104106149530825, 14.70061816801487348185576092836, 15.83357387720452912021821819094, 16.832209737533765087696602451790, 17.42887729235443596540648224445, 18.04368396306681444170415399104, 18.97739400157934601552786722400, 19.796843270849072706966407217863, 20.68127323464345004042331417571, 21.232246289655230812346467877322, 21.94053680556388626598749826520

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