Dirichlet series
L(s) = 1 | + (0.983 − 0.181i)2-s + (0.990 + 0.136i)3-s + (0.934 − 0.356i)4-s + (0.949 + 0.313i)5-s + (0.998 − 0.0455i)6-s + (−0.803 + 0.595i)7-s + (0.854 − 0.519i)8-s + (0.962 + 0.269i)9-s + (0.990 + 0.136i)10-s + (0.962 + 0.269i)11-s + (0.974 − 0.225i)12-s + (0.247 − 0.968i)13-s + (−0.682 + 0.730i)14-s + (0.898 + 0.439i)15-s + (0.746 − 0.665i)16-s + (−0.0682 + 0.997i)17-s + ⋯ |
L(s) = 1 | + (0.983 − 0.181i)2-s + (0.990 + 0.136i)3-s + (0.934 − 0.356i)4-s + (0.949 + 0.313i)5-s + (0.998 − 0.0455i)6-s + (−0.803 + 0.595i)7-s + (0.854 − 0.519i)8-s + (0.962 + 0.269i)9-s + (0.990 + 0.136i)10-s + (0.962 + 0.269i)11-s + (0.974 − 0.225i)12-s + (0.247 − 0.968i)13-s + (−0.682 + 0.730i)14-s + (0.898 + 0.439i)15-s + (0.746 − 0.665i)16-s + (−0.0682 + 0.997i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(967\) |
Sign: | $0.972 + 0.232i$ |
Analytic conductor: | \(103.918\) |
Root analytic conductor: | \(103.918\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{967} (567, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 967,\ (1:\ ),\ 0.972 + 0.232i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(8.250128271 + 0.9719790836i\) |
\(L(\frac12)\) | \(\approx\) | \(8.250128271 + 0.9719790836i\) |
\(L(1)\) | \(\approx\) | \(3.342611905 + 0.1196256230i\) |
\(L(1)\) | \(\approx\) | \(3.342611905 + 0.1196256230i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.983 - 0.181i)T \) |
3 | \( 1 + (0.990 + 0.136i)T \) | |
5 | \( 1 + (0.949 + 0.313i)T \) | |
7 | \( 1 + (-0.803 + 0.595i)T \) | |
11 | \( 1 + (0.962 + 0.269i)T \) | |
13 | \( 1 + (0.247 - 0.968i)T \) | |
17 | \( 1 + (-0.0682 + 0.997i)T \) | |
19 | \( 1 + (-0.377 - 0.926i)T \) | |
23 | \( 1 + (0.775 + 0.631i)T \) | |
29 | \( 1 + (-0.962 + 0.269i)T \) | |
31 | \( 1 + (0.0227 + 0.999i)T \) | |
37 | \( 1 + (0.247 - 0.968i)T \) | |
41 | \( 1 + (-0.460 + 0.887i)T \) | |
43 | \( 1 + (0.829 - 0.557i)T \) | |
47 | \( 1 + (-0.898 - 0.439i)T \) | |
53 | \( 1 + (-0.158 + 0.987i)T \) | |
59 | \( 1 + (-0.158 - 0.987i)T \) | |
61 | \( 1 + (0.538 + 0.842i)T \) | |
67 | \( 1 + (-0.854 - 0.519i)T \) | |
71 | \( 1 + (-0.334 - 0.942i)T \) | |
73 | \( 1 + (-0.158 - 0.987i)T \) | |
79 | \( 1 + (0.158 + 0.987i)T \) | |
83 | \( 1 + (-0.998 + 0.0455i)T \) | |
89 | \( 1 + (0.877 - 0.480i)T \) | |
97 | \( 1 + T \) | |
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Imaginary part of the first few zeros on the critical line
−21.46560446287827289866458622806, −20.59447182786386421542694447439, −20.44517261703270887713295389133, −19.23799746232925723826578563163, −18.736148143338047699427180891769, −17.255868172840889113141094678282, −16.569348316375875939669972162870, −16.05226966037553883493940263217, −14.73958244637682760472908589249, −14.31540352785413662918800873803, −13.51133845496068814332041721151, −13.14595513348040467925766625901, −12.23806417095472327317331698931, −11.21302458793301330570392999466, −10.013504288855504465529074236, −9.37412966465672269768975084098, −8.507702712039397675248528722899, −7.2756517994914133918341005777, −6.62469307201197910559944519807, −5.9568029584386903737761053249, −4.58141301259951567843424252267, −3.90898431115614115133206208394, −2.99961126552467666180439283750, −2.049186397286360242587374365878, −1.13769606212938297036329733607, 1.35078152455216379400834268386, 2.18928923105627007600298398531, 3.094605023342991602667840922540, 3.63659848687059123562565921668, 4.87135170898149384511284985351, 5.85443625734878362856845592321, 6.590949882464696443978844795444, 7.38497312182070788982447277759, 8.80078443301556917564811920548, 9.42444484139960895139377028555, 10.30428789243394748995966828408, 11.02512920828201257270282778837, 12.39609699876911100545962357986, 13.026321399702077056234053875040, 13.46483333132054491167878633860, 14.5254094353949216520951216809, 15.00072920995940904956988405074, 15.62251307106896297439189050024, 16.67416521580983637788103055008, 17.65911598128917496841569945874, 18.74043963236839129119303621285, 19.60000698293632368087822093815, 19.933376401110276471817548201486, 21.04283444662386489700726996497, 21.61161620918054111805827569678