Properties

Label 1-967-967.454-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.652 + 0.757i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.998 − 0.0455i)2-s + (−0.682 + 0.730i)3-s + (0.995 + 0.0909i)4-s + (0.648 − 0.761i)5-s + (0.715 − 0.699i)6-s + (0.158 − 0.987i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (−0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (−0.746 + 0.665i)12-s + (−0.898 + 0.439i)13-s + (−0.203 + 0.979i)14-s + (0.113 + 0.993i)15-s + (0.983 + 0.181i)16-s + (−0.917 + 0.398i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0455i)2-s + (−0.682 + 0.730i)3-s + (0.995 + 0.0909i)4-s + (0.648 − 0.761i)5-s + (0.715 − 0.699i)6-s + (0.158 − 0.987i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (−0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (−0.746 + 0.665i)12-s + (−0.898 + 0.439i)13-s + (−0.203 + 0.979i)14-s + (0.113 + 0.993i)15-s + (0.983 + 0.181i)16-s + (−0.917 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.652 + 0.757i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.652 + 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09778490393 - 0.2132409771i\)
\(L(\frac12)\) \(\approx\) \(-0.09778490393 - 0.2132409771i\)
\(L(1)\) \(\approx\) \(0.4977032337 - 0.1417404585i\)
\(L(1)\) \(\approx\) \(0.4977032337 - 0.1417404585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.998 + 0.0455i)T \)
3 \( 1 + (0.682 - 0.730i)T \)
5 \( 1 + (-0.648 + 0.761i)T \)
7 \( 1 + (-0.158 + 0.987i)T \)
11 \( 1 + (0.0682 + 0.997i)T \)
13 \( 1 + (0.898 - 0.439i)T \)
17 \( 1 + (0.917 - 0.398i)T \)
19 \( 1 + (0.291 + 0.956i)T \)
23 \( 1 + (-0.576 + 0.816i)T \)
29 \( 1 + (-0.0682 + 0.997i)T \)
31 \( 1 + (-0.377 - 0.926i)T \)
37 \( 1 + (0.898 - 0.439i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (0.803 - 0.595i)T \)
47 \( 1 + (0.113 + 0.993i)T \)
53 \( 1 + (0.419 + 0.907i)T \)
59 \( 1 + (0.419 - 0.907i)T \)
61 \( 1 + (0.247 + 0.968i)T \)
67 \( 1 + (-0.990 + 0.136i)T \)
71 \( 1 + (-0.460 + 0.887i)T \)
73 \( 1 + (0.419 - 0.907i)T \)
79 \( 1 + (-0.419 + 0.907i)T \)
83 \( 1 + (0.715 - 0.699i)T \)
89 \( 1 + (0.613 + 0.789i)T \)
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

−22.11970146115025689557510736845, −21.30163099807339349444178957003, −20.28021076420196981694891665880, −19.380742014993909476862739425895, −18.598784240552425034917951573843, −18.19985760921316411239183590321, −17.38683984744838094158248734876, −17.08911924639057368624087347800, −15.73792262700221607749322870435, −15.11123980522685562211667459607, −14.28590482254669422413780545224, −12.997858622579067670535217582184, −12.26399473751414772760268319045, −11.55303920398797392440332044159, −10.70863463432814547296380877475, −9.95399498252704216503534260253, −9.176711687638555456994067788758, −8.04966376546479479349834459186, −7.22046581569104806073675819875, −6.64268341498871156099916190722, −5.72859226830841625573988336828, −5.02075259137710330458110366472, −2.957919037157502277664398213763, −2.14059313888667730847963228778, −1.586565099995736064915571755350, 0.10076846088961052209890683754, 0.72761265523965647553564134947, 1.92079812252587772577098261696, 3.21455279922698429327968405921, 4.47744814280498702510484080449, 5.16465984001561198318023956799, 6.43531711846286972206207385730, 6.831292227018773403175361978038, 8.33547913706572739727910965642, 8.88171273682561448087019702076, 9.81140252727972608560434746263, 10.41686944214758578801680939717, 11.13844473465744114979873138226, 11.90314597350634816601962098659, 12.9380386683106419959333646555, 13.88199522140723310112928896326, 14.99010310786623613236168951770, 15.876764912270331658072144722604, 16.611689974888509174046600150686, 17.23438443077947647714471529786, 17.417527853871354372054678236, 18.55737691699188375801922533274, 19.675696632289269506542098741670, 20.15175573962769250977972341765, 21.238260381060615784617873237263

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