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: | $\chi_{967} (354, \cdot )$ |
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
−22.14102297715600056879608849191, −21.613596545386656096536633478722, −20.50633381205270547679462411572, −19.73330143917133919947986569005, −18.539382666724482705558344780191, −17.90383418842154866303661905992, −16.79230731494853343475957106945, −16.394667809069920463481576254972, −15.62573757334526298145203088723, −15.3456249590287106322526448781, −13.87711509359678407138349811460, −13.1543491251109288170707465263, −12.11764538641507739922298175427, −12.03564652397498506179820870313, −10.9776712077508542123725950375, −9.65048866234166355868444309067, −8.88945013325804319707659922761, −7.85124971691178959517105747637, −7.04897717944225694112993354537, −6.03260583222291131105229901404, −5.45283345375596531830799668114, −4.69787222993813766297588205675, −3.65872564565486467063763270272, −2.89916928894000403123301557719, −0.74158689142797656296553117682, 0.0311420483963184190224438696, 1.31100799741656892927476042975, 2.34316885509492543435976353699, 3.842540454599911435004901386075, 4.01948375712260817520274807094, 5.21281500864460931186902900458, 6.36087919141082195115991792534, 6.79086609702624271378941496494, 7.71671979689052655513335554530, 9.42292442163953875873408341998, 10.18361580724480850732590556129, 10.70110781305075264014843006563, 11.694579325245462405199937735257, 12.2393736146687314191328180623, 12.79432411105683672861118267750, 13.96875771978343627233560041895, 14.58190457331944546927377865656, 15.68019595794886541317772967693, 16.230519148061797873830584738344, 17.28244810702313918223646279340, 18.27635942431689214365989482395, 18.84132960286905679469478097701, 19.75066725602999786623580741228, 20.08343165852744183233942647835, 21.564175397340494678160484646712