Dirichlet series
L(s) = 1 | + (0.981 − 0.193i)2-s + (0.864 − 0.502i)3-s + (0.924 − 0.380i)4-s + (−0.993 − 0.116i)5-s + (0.750 − 0.660i)6-s + (−0.00975 − 0.999i)7-s + (0.833 − 0.552i)8-s + (0.494 − 0.869i)9-s + (−0.996 + 0.0779i)10-s + (−0.957 − 0.288i)11-s + (0.608 − 0.793i)12-s + (0.957 − 0.288i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.710 − 0.703i)16-s + (−0.184 + 0.982i)17-s + ⋯ |
L(s) = 1 | + (0.981 − 0.193i)2-s + (0.864 − 0.502i)3-s + (0.924 − 0.380i)4-s + (−0.993 − 0.116i)5-s + (0.750 − 0.660i)6-s + (−0.00975 − 0.999i)7-s + (0.833 − 0.552i)8-s + (0.494 − 0.869i)9-s + (−0.996 + 0.0779i)10-s + (−0.957 − 0.288i)11-s + (0.608 − 0.793i)12-s + (0.957 − 0.288i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.710 − 0.703i)16-s + (−0.184 + 0.982i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(967\) |
Sign: | $-0.974 - 0.225i$ |
Analytic conductor: | \(103.918\) |
Root analytic conductor: | \(103.918\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{967} (191, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 967,\ (1:\ ),\ -0.974 - 0.225i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.4800240799 - 4.194765137i\) |
\(L(\frac12)\) | \(\approx\) | \(0.4800240799 - 4.194765137i\) |
\(L(1)\) | \(\approx\) | \(1.723571477 - 1.294989529i\) |
\(L(1)\) | \(\approx\) | \(1.723571477 - 1.294989529i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.981 - 0.193i)T \) |
3 | \( 1 + (0.864 - 0.502i)T \) | |
5 | \( 1 + (-0.993 - 0.116i)T \) | |
7 | \( 1 + (-0.00975 - 0.999i)T \) | |
11 | \( 1 + (-0.957 - 0.288i)T \) | |
13 | \( 1 + (0.957 - 0.288i)T \) | |
17 | \( 1 + (-0.184 + 0.982i)T \) | |
19 | \( 1 + (0.145 - 0.989i)T \) | |
23 | \( 1 + (0.668 - 0.744i)T \) | |
29 | \( 1 + (-0.987 - 0.155i)T \) | |
31 | \( 1 + (0.353 + 0.935i)T \) | |
37 | \( 1 + (0.822 + 0.568i)T \) | |
41 | \( 1 + (-0.962 + 0.269i)T \) | |
43 | \( 1 + (-0.592 - 0.805i)T \) | |
47 | \( 1 + (0.260 - 0.965i)T \) | |
53 | \( 1 + (-0.576 - 0.816i)T \) | |
59 | \( 1 + (-0.998 + 0.0585i)T \) | |
61 | \( 1 + (0.962 - 0.269i)T \) | |
67 | \( 1 + (-0.353 + 0.935i)T \) | |
71 | \( 1 + (0.981 + 0.193i)T \) | |
73 | \( 1 + (-0.576 + 0.816i)T \) | |
79 | \( 1 + (-0.874 + 0.485i)T \) | |
83 | \( 1 + (-0.477 - 0.878i)T \) | |
89 | \( 1 + (-0.353 + 0.935i)T \) | |
97 | \( 1 + (-0.900 + 0.433i)T \) | |
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Imaginary part of the first few zeros on the critical line
−21.94300561126761120553352645809, −20.911032493275491953542402442, −20.73283827005292374783435366177, −19.809402297148881036329682502853, −18.8173122736239482503264380454, −18.37695527650134873046082787393, −16.64664490064126566911380887273, −15.96576088454794836300618312352, −15.438165901208564939007679152684, −14.97844087789163337033936044977, −14.06237264413544446020181236733, −13.20555372212785356336298432404, −12.51088382291060660342627389155, −11.45990194100689258667644360320, −10.986530156049543916421438315107, −9.72559286555418700122441967653, −8.72202347439648157685371287029, −7.861640454600049760353267148429, −7.371094027891787333819318643, −6.03819782722306771845264297027, −5.09874669669522208370383015890, −4.326650637014457068815024043867, −3.363060302591865344789839077004, −2.809061263753833599837130387325, −1.73808337569615931803071948655, 0.51481954853230322373623631327, 1.45296219775822510187283498687, 2.78648803354948611595027635008, 3.494421251654302457598956614852, 4.16036254358882637399114250859, 5.14293145271694950907513237026, 6.53328103982539736986808882342, 7.10768328089588079664942651066, 8.02145686999165250484258116782, 8.610341063695575254278432294431, 10.15181386930226380552090307282, 10.87998834298384091147841764831, 11.60050515394751875794264990814, 12.8814061049202908616764669743, 13.06464611160194114674888427296, 13.83743056671122828751725842238, 14.83967387788298832556671969538, 15.403052392391735085278490531524, 16.09125009339279456618352361585, 17.086258525354840370545962120985, 18.44011468911219123906032940401, 19.05775463371899318142137289615, 19.9315320783307604947047462812, 20.338162838113361372544155903070, 20.95877186584419733064431080403