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
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.998 - 0.0455i)T \) |
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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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