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Properties

Label 1-967-967.539-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.571 - 0.820i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.419 − 0.907i)2-s + (−0.0682 − 0.997i)3-s + (−0.648 + 0.761i)4-s + (0.934 + 0.356i)5-s + (−0.877 + 0.480i)6-s + (0.746 − 0.665i)7-s + (0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (−0.0682 − 0.997i)10-s + (−0.990 + 0.136i)11-s + (0.803 + 0.595i)12-s + (0.377 + 0.926i)13-s + (−0.917 − 0.398i)14-s + (0.291 − 0.956i)15-s + (−0.158 − 0.987i)16-s + (0.682 − 0.730i)17-s + ⋯
L(s)  = 1  + (−0.419 − 0.907i)2-s + (−0.0682 − 0.997i)3-s + (−0.648 + 0.761i)4-s + (0.934 + 0.356i)5-s + (−0.877 + 0.480i)6-s + (0.746 − 0.665i)7-s + (0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (−0.0682 − 0.997i)10-s + (−0.990 + 0.136i)11-s + (0.803 + 0.595i)12-s + (0.377 + 0.926i)13-s + (−0.917 − 0.398i)14-s + (0.291 − 0.956i)15-s + (−0.158 − 0.987i)16-s + (0.682 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.571 - 0.820i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.571 - 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6228606229 - 1.193620236i\)
\(L(\frac12)\) \(\approx\) \(0.6228606229 - 1.193620236i\)
\(L(1)\) \(\approx\) \(0.7636628870 - 0.6480931488i\)
\(L(1)\) \(\approx\) \(0.7636628870 - 0.6480931488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.419 - 0.907i)T \)
3 \( 1 + (-0.0682 - 0.997i)T \)
5 \( 1 + (0.934 + 0.356i)T \)
7 \( 1 + (0.746 - 0.665i)T \)
11 \( 1 + (-0.990 + 0.136i)T \)
13 \( 1 + (0.377 + 0.926i)T \)
17 \( 1 + (0.682 - 0.730i)T \)
19 \( 1 + (0.898 + 0.439i)T \)
23 \( 1 + (-0.334 - 0.942i)T \)
29 \( 1 + (-0.990 - 0.136i)T \)
31 \( 1 + (-0.247 - 0.968i)T \)
37 \( 1 + (0.377 + 0.926i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (-0.974 + 0.225i)T \)
47 \( 1 + (0.291 - 0.956i)T \)
53 \( 1 + (0.983 + 0.181i)T \)
59 \( 1 + (0.983 - 0.181i)T \)
61 \( 1 + (0.0227 - 0.999i)T \)
67 \( 1 + (0.962 - 0.269i)T \)
71 \( 1 + (-0.576 - 0.816i)T \)
73 \( 1 + (0.983 - 0.181i)T \)
79 \( 1 + (0.983 - 0.181i)T \)
83 \( 1 + (-0.877 + 0.480i)T \)
89 \( 1 + (-0.715 - 0.699i)T \)
97 \( 1 + T \)
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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

−21.92428259078852745487290560446, −21.33513004264302168000320734406, −20.61328075040556090511555359536, −19.731671167250329033296348067995, −18.460513263629615185974503287653, −17.82450516294219494233148697519, −17.39798220200197234387226084307, −16.3475994957329333612439705382, −15.800047193860734198481539601686, −15.04385167175817427046849602920, −14.35470219239039159633247001307, −13.527050978523153333463152372858, −12.6171557954453143860698983105, −11.21290990360260849188837005325, −10.46303832424492714577200063428, −9.78383079737443115487027861201, −8.970372651707377960407983575888, −8.315822662137263252070517338114, −7.48737809794360597858112495369, −5.86940871979568569102827262616, −5.48833691574790281365374003368, −5.0807049919932153493747207468, −3.733856963291182630594453301173, −2.42776140872303333989111890556, −1.12036043322302267166693377170, 0.80918608130557373465914934709, 1.79438967982029375236558687414, 2.39715326029060675722402008307, 3.46731747287535940104170629864, 4.809188392060981035203074859901, 5.67434498820369927717183505832, 6.91607661291705242644163626261, 7.65387596620064012452096505147, 8.34714984554236471729573510893, 9.49707609167900150419263444314, 10.19210871367103879955479572638, 11.16149593937074526469151810397, 11.64397444336532406838095458566, 12.70613822395542555202370428057, 13.51662944310564508572632112067, 13.9023691085628424069870236062, 14.68675958972846538080602701164, 16.590930807339293221001607546155, 16.8548222027502248304162033323, 18.014783771179463139357945286455, 18.39205271540535156907934685606, 18.7260982710494774767363041853, 20.0515534551605518457264452216, 20.67147156120382290945464289059, 21.14208103432209547070288955316

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