Dirichlet series
L(s) = 1 | + (0.165 + 0.986i)2-s + (0.822 − 0.568i)3-s + (−0.945 + 0.325i)4-s + (0.977 − 0.212i)5-s + (0.696 + 0.717i)6-s + (−0.241 − 0.970i)7-s + (−0.477 − 0.878i)8-s + (0.353 − 0.935i)9-s + (0.371 + 0.928i)10-s + (−0.511 − 0.859i)11-s + (−0.592 + 0.805i)12-s + (0.511 − 0.859i)13-s + (0.917 − 0.398i)14-s + (0.682 − 0.730i)15-s + (0.787 − 0.615i)16-s + (0.996 − 0.0779i)17-s + ⋯ |
L(s) = 1 | + (0.165 + 0.986i)2-s + (0.822 − 0.568i)3-s + (−0.945 + 0.325i)4-s + (0.977 − 0.212i)5-s + (0.696 + 0.717i)6-s + (−0.241 − 0.970i)7-s + (−0.477 − 0.878i)8-s + (0.353 − 0.935i)9-s + (0.371 + 0.928i)10-s + (−0.511 − 0.859i)11-s + (−0.592 + 0.805i)12-s + (0.511 − 0.859i)13-s + (0.917 − 0.398i)14-s + (0.682 − 0.730i)15-s + (0.787 − 0.615i)16-s + (0.996 − 0.0779i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(967\) |
Sign: | $0.0546 - 0.998i$ |
Analytic conductor: | \(103.918\) |
Root analytic conductor: | \(103.918\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{967} (109, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 967,\ (1:\ ),\ 0.0546 - 0.998i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(2.103780828 - 1.991703415i\) |
\(L(\frac12)\) | \(\approx\) | \(2.103780828 - 1.991703415i\) |
\(L(1)\) | \(\approx\) | \(1.540831750 - 0.1313875026i\) |
\(L(1)\) | \(\approx\) | \(1.540831750 - 0.1313875026i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.165 + 0.986i)T \) |
3 | \( 1 + (0.822 - 0.568i)T \) | |
5 | \( 1 + (0.977 - 0.212i)T \) | |
7 | \( 1 + (-0.241 - 0.970i)T \) | |
11 | \( 1 + (-0.511 - 0.859i)T \) | |
13 | \( 1 + (0.511 - 0.859i)T \) | |
17 | \( 1 + (0.996 - 0.0779i)T \) | |
19 | \( 1 + (-0.494 + 0.869i)T \) | |
23 | \( 1 + (-0.527 - 0.849i)T \) | |
29 | \( 1 + (0.724 + 0.689i)T \) | |
31 | \( 1 + (0.389 - 0.921i)T \) | |
37 | \( 1 + (-0.833 + 0.552i)T \) | |
41 | \( 1 + (-0.854 + 0.519i)T \) | |
43 | \( 1 + (0.145 + 0.989i)T \) | |
47 | \( 1 + (0.297 - 0.954i)T \) | |
53 | \( 1 + (-0.334 + 0.942i)T \) | |
59 | \( 1 + (-0.107 + 0.994i)T \) | |
61 | \( 1 + (0.854 - 0.519i)T \) | |
67 | \( 1 + (-0.389 - 0.921i)T \) | |
71 | \( 1 + (0.165 - 0.986i)T \) | |
73 | \( 1 + (-0.334 - 0.942i)T \) | |
79 | \( 1 + (-0.993 + 0.116i)T \) | |
83 | \( 1 + (0.126 - 0.991i)T \) | |
89 | \( 1 + (-0.389 - 0.921i)T \) | |
97 | \( 1 + (-0.222 - 0.974i)T \) | |
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Imaginary part of the first few zeros on the critical line
−21.60776494052006613914659921242, −21.00182064549822154857290986836, −20.48309163011689181377664764864, −19.17790547381413828525312106173, −19.05032976797256940503208632228, −17.963096809094251021370929271095, −17.329453293877304032161871731732, −15.95420122712850786279887820502, −15.26543622248946077803483994923, −14.324022393231554291648903142504, −13.83032032057733418561278117036, −12.98314744085851892803837646154, −12.22751984355689264783399065984, −11.1876572819461048161096719126, −10.13585634744740212353955702612, −9.80972621453179707460511388599, −8.96275699000404454706425916604, −8.36431089465163720773784303281, −6.91160920653025806791617648948, −5.631985601471789755003635042, −5.00776143461579697187851524618, −3.94138543095324765433250954035, −2.92962448750724433331560378923, −2.250209925948735141943568828881, −1.56714042896349959840228464856, 0.49536045232031759416227584656, 1.35624115868500908188301749537, 2.932929760450441560457433659914, 3.5970580626812033701795665786, 4.79443305209934646485544662162, 5.99223507968579550143696873459, 6.35433504260974616986985174190, 7.52642865782309314272527204792, 8.189194304386387303261006067156, 8.800360206830250617232105874400, 10.026288173207634741728158955111, 10.35550675992115891049766694788, 12.23788357489859520052683097966, 13.01248070686246551071004257173, 13.548478003640084864636337528694, 14.11583954816446474195685082969, 14.77563475794544829731898153991, 15.891287768037282940245231574, 16.66219294282268895514726967742, 17.28274931932744167286002204773, 18.381891411746204118083053952, 18.576488109557632384156419122844, 19.79306793712846930341587487206, 20.7963334396585327705817281291, 21.1594249787490652113312588523