Dirichlet series
L(s) = 1 | + (0.811 + 0.584i)2-s + (−0.909 − 0.416i)3-s + (0.316 + 0.948i)4-s + (0.967 + 0.250i)5-s + (−0.494 − 0.869i)6-s + (−0.279 − 0.960i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (0.638 + 0.769i)10-s + (0.592 − 0.805i)11-s + (0.107 − 0.994i)12-s + (−0.592 − 0.805i)13-s + (0.334 − 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.799 + 0.600i)16-s + (0.787 + 0.615i)17-s + ⋯ |
L(s) = 1 | + (0.811 + 0.584i)2-s + (−0.909 − 0.416i)3-s + (0.316 + 0.948i)4-s + (0.967 + 0.250i)5-s + (−0.494 − 0.869i)6-s + (−0.279 − 0.960i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (0.638 + 0.769i)10-s + (0.592 − 0.805i)11-s + (0.107 − 0.994i)12-s + (−0.592 − 0.805i)13-s + (0.334 − 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.799 + 0.600i)16-s + (0.787 + 0.615i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(967\) |
Sign: | $0.887 + 0.460i$ |
Analytic conductor: | \(103.918\) |
Root analytic conductor: | \(103.918\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{967} (720, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 967,\ (1:\ ),\ 0.887 + 0.460i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(3.035642201 + 0.7401390503i\) |
\(L(\frac12)\) | \(\approx\) | \(3.035642201 + 0.7401390503i\) |
\(L(1)\) | \(\approx\) | \(1.509296936 + 0.3227376137i\) |
\(L(1)\) | \(\approx\) | \(1.509296936 + 0.3227376137i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.811 + 0.584i)T \) |
3 | \( 1 + (-0.909 - 0.416i)T \) | |
5 | \( 1 + (0.967 + 0.250i)T \) | |
7 | \( 1 + (-0.279 - 0.960i)T \) | |
11 | \( 1 + (0.592 - 0.805i)T \) | |
13 | \( 1 + (-0.592 - 0.805i)T \) | |
17 | \( 1 + (0.787 + 0.615i)T \) | |
19 | \( 1 + (-0.892 + 0.451i)T \) | |
23 | \( 1 + (0.696 + 0.717i)T \) | |
29 | \( 1 + (0.184 + 0.982i)T \) | |
31 | \( 1 + (-0.864 - 0.502i)T \) | |
37 | \( 1 + (0.260 - 0.965i)T \) | |
41 | \( 1 + (0.0682 + 0.997i)T \) | |
43 | \( 1 + (0.442 - 0.896i)T \) | |
47 | \( 1 + (0.977 - 0.212i)T \) | |
53 | \( 1 + (0.854 - 0.519i)T \) | |
59 | \( 1 + (0.126 + 0.991i)T \) | |
61 | \( 1 + (-0.0682 - 0.997i)T \) | |
67 | \( 1 + (0.864 - 0.502i)T \) | |
71 | \( 1 + (0.811 - 0.584i)T \) | |
73 | \( 1 + (0.854 + 0.519i)T \) | |
79 | \( 1 + (0.544 + 0.838i)T \) | |
83 | \( 1 + (-0.957 + 0.288i)T \) | |
89 | \( 1 + (0.864 - 0.502i)T \) | |
97 | \( 1 + (-0.900 + 0.433i)T \) | |
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Imaginary part of the first few zeros on the critical line
−21.49919353638867874618545865673, −21.1069704577127374045848430285, −20.23302272370061165966862479024, −19.056540095287946461627831775420, −18.494627805828846288794662488163, −17.46835029505660855146433752901, −16.7763192982967301498101556583, −15.90821041096006038916419200429, −14.94339029931390525764570319335, −14.45714347472204902007366838407, −13.30903214417009329986317547872, −12.36891750006558861813067575258, −12.16258659802895491993746504239, −11.16316817305411090938866432700, −10.21946160057524037532785570717, −9.503306402935474944623465780821, −9.06120169882037437633590013472, −6.938907918720162504142077278157, −6.40525398804724485482010149834, −5.521159646552150547354507891728, −4.87705285492379024429225106731, −4.133508050841829267975547723974, −2.70651302007363628526099071913, −1.94363334317684031259352799397, −0.79464622202152751438195655932, 0.75578984929596688243388813545, 1.931428673572676501038683634664, 3.22998425289714351303747411758, 4.11644180782345421504202144790, 5.40022762601196891137154654986, 5.75261627837481852411938374352, 6.68683887833894668480624266720, 7.25960062562990511499398780107, 8.22713320763778583893493704989, 9.56676808838976427144042672223, 10.659975384893615020469040669048, 11.05083795087089264208189111545, 12.4038157706821985990884865397, 12.83732520983582601547866461071, 13.63198482354721412664700110979, 14.31378847098767173830170746703, 15.130230338251118235298367617143, 16.45567374626087125811784229686, 16.85767090257539350008825969695, 17.31107495035967719314985917677, 18.17481002865901218344038352887, 19.21861721703517208803873515536, 20.14181290103154730426562412485, 21.372283593233666570078049075715, 21.66978139618407867161641422239