Dirichlet series
L(s) = 1 | + (0.653 + 0.756i)2-s + (−0.981 + 0.193i)3-s + (−0.145 + 0.989i)4-s + (−0.737 − 0.675i)5-s + (−0.787 − 0.615i)6-s + (−0.833 + 0.552i)7-s + (−0.844 + 0.536i)8-s + (0.924 − 0.380i)9-s + (0.0292 − 0.999i)10-s + (0.279 − 0.960i)11-s + (−0.0487 − 0.998i)12-s + (−0.279 − 0.960i)13-s + (−0.962 − 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.957 − 0.288i)16-s + (0.126 − 0.991i)17-s + ⋯ |
L(s) = 1 | + (0.653 + 0.756i)2-s + (−0.981 + 0.193i)3-s + (−0.145 + 0.989i)4-s + (−0.737 − 0.675i)5-s + (−0.787 − 0.615i)6-s + (−0.833 + 0.552i)7-s + (−0.844 + 0.536i)8-s + (0.924 − 0.380i)9-s + (0.0292 − 0.999i)10-s + (0.279 − 0.960i)11-s + (−0.0487 − 0.998i)12-s + (−0.279 − 0.960i)13-s + (−0.962 − 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.957 − 0.288i)16-s + (0.126 − 0.991i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(967\) |
Sign: | $-0.993 - 0.116i$ |
Analytic conductor: | \(103.918\) |
Root analytic conductor: | \(103.918\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{967} (112, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 967,\ (1:\ ),\ -0.993 - 0.116i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.006222584133 + 0.1065880942i\) |
\(L(\frac12)\) | \(\approx\) | \(0.006222584133 + 0.1065880942i\) |
\(L(1)\) | \(\approx\) | \(0.7067459729 + 0.2157471529i\) |
\(L(1)\) | \(\approx\) | \(0.7067459729 + 0.2157471529i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.653 + 0.756i)T \) |
3 | \( 1 + (-0.981 + 0.193i)T \) | |
5 | \( 1 + (-0.737 - 0.675i)T \) | |
7 | \( 1 + (-0.833 + 0.552i)T \) | |
11 | \( 1 + (0.279 - 0.960i)T \) | |
13 | \( 1 + (-0.279 - 0.960i)T \) | |
17 | \( 1 + (0.126 - 0.991i)T \) | |
19 | \( 1 + (0.799 - 0.600i)T \) | |
23 | \( 1 + (-0.996 + 0.0779i)T \) | |
29 | \( 1 + (0.998 - 0.0585i)T \) | |
31 | \( 1 + (-0.945 - 0.325i)T \) | |
37 | \( 1 + (-0.165 + 0.986i)T \) | |
41 | \( 1 + (0.775 - 0.631i)T \) | |
43 | \( 1 + (-0.938 + 0.344i)T \) | |
47 | \( 1 + (0.995 + 0.0974i)T \) | |
53 | \( 1 + (0.682 + 0.730i)T \) | |
59 | \( 1 + (-0.932 + 0.362i)T \) | |
61 | \( 1 + (-0.775 + 0.631i)T \) | |
67 | \( 1 + (0.945 - 0.325i)T \) | |
71 | \( 1 + (0.653 - 0.756i)T \) | |
73 | \( 1 + (0.682 - 0.730i)T \) | |
79 | \( 1 + (-0.560 - 0.828i)T \) | |
83 | \( 1 + (0.00975 + 0.999i)T \) | |
89 | \( 1 + (0.945 - 0.325i)T \) | |
97 | \( 1 + (-0.222 - 0.974i)T \) | |
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Imaginary part of the first few zeros on the critical line
−21.564175397340494678160484646712, −20.08343165852744183233942647835, −19.75066725602999786623580741228, −18.84132960286905679469478097701, −18.27635942431689214365989482395, −17.28244810702313918223646279340, −16.230519148061797873830584738344, −15.68019595794886541317772967693, −14.58190457331944546927377865656, −13.96875771978343627233560041895, −12.79432411105683672861118267750, −12.2393736146687314191328180623, −11.694579325245462405199937735257, −10.70110781305075264014843006563, −10.18361580724480850732590556129, −9.42292442163953875873408341998, −7.71671979689052655513335554530, −6.79086609702624271378941496494, −6.36087919141082195115991792534, −5.21281500864460931186902900458, −4.01948375712260817520274807094, −3.842540454599911435004901386075, −2.34316885509492543435976353699, −1.31100799741656892927476042975, −0.0311420483963184190224438696, 0.74158689142797656296553117682, 2.89916928894000403123301557719, 3.65872564565486467063763270272, 4.69787222993813766297588205675, 5.45283345375596531830799668114, 6.03260583222291131105229901404, 7.04897717944225694112993354537, 7.85124971691178959517105747637, 8.88945013325804319707659922761, 9.65048866234166355868444309067, 10.9776712077508542123725950375, 12.03564652397498506179820870313, 12.11764538641507739922298175427, 13.1543491251109288170707465263, 13.87711509359678407138349811460, 15.3456249590287106322526448781, 15.62573757334526298145203088723, 16.394667809069920463481576254972, 16.79230731494853343475957106945, 17.90383418842154866303661905992, 18.539382666724482705558344780191, 19.73330143917133919947986569005, 20.50633381205270547679462411572, 21.613596545386656096536633478722, 22.14102297715600056879608849191