Dirichlet series
L(s) = 1 | + (0.195 − 0.980i)2-s + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)4-s + (0.634 + 0.773i)5-s + (−0.831 − 0.555i)6-s + (−0.707 − 0.707i)7-s + (−0.555 + 0.831i)8-s + (−0.707 − 0.707i)9-s + (0.881 − 0.471i)10-s + (−0.0980 − 0.995i)11-s + (−0.707 + 0.707i)12-s + (0.471 − 0.881i)13-s + (−0.831 + 0.555i)14-s + (0.956 − 0.290i)15-s + (0.707 + 0.707i)16-s + (−0.634 + 0.773i)17-s + ⋯ |
L(s) = 1 | + (0.195 − 0.980i)2-s + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)4-s + (0.634 + 0.773i)5-s + (−0.831 − 0.555i)6-s + (−0.707 − 0.707i)7-s + (−0.555 + 0.831i)8-s + (−0.707 − 0.707i)9-s + (0.881 − 0.471i)10-s + (−0.0980 − 0.995i)11-s + (−0.707 + 0.707i)12-s + (0.471 − 0.881i)13-s + (−0.831 + 0.555i)14-s + (0.956 − 0.290i)15-s + (0.707 + 0.707i)16-s + (−0.634 + 0.773i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(193\) |
Sign: | $-0.336 + 0.941i$ |
Analytic conductor: | \(20.7407\) |
Root analytic conductor: | \(20.7407\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{193} (164, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 193,\ (1:\ ),\ -0.336 + 0.941i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(-0.5435109522 - 0.7718104182i\) |
\(L(\frac12)\) | \(\approx\) | \(-0.5435109522 - 0.7718104182i\) |
\(L(1)\) | \(\approx\) | \(0.5841910041 - 0.7948053179i\) |
\(L(1)\) | \(\approx\) | \(0.5841910041 - 0.7948053179i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 193 | \( 1 \) |
good | 2 | \( 1 + (0.195 - 0.980i)T \) |
3 | \( 1 + (0.382 - 0.923i)T \) | |
5 | \( 1 + (0.634 + 0.773i)T \) | |
7 | \( 1 + (-0.707 - 0.707i)T \) | |
11 | \( 1 + (-0.0980 - 0.995i)T \) | |
13 | \( 1 + (0.471 - 0.881i)T \) | |
17 | \( 1 + (-0.634 + 0.773i)T \) | |
19 | \( 1 + (-0.773 + 0.634i)T \) | |
23 | \( 1 + (0.195 - 0.980i)T \) | |
29 | \( 1 + (-0.290 + 0.956i)T \) | |
31 | \( 1 + (-0.980 - 0.195i)T \) | |
37 | \( 1 + (-0.290 - 0.956i)T \) | |
41 | \( 1 + (0.0980 + 0.995i)T \) | |
43 | \( 1 + (-0.707 + 0.707i)T \) | |
47 | \( 1 + (0.881 + 0.471i)T \) | |
53 | \( 1 + (0.471 - 0.881i)T \) | |
59 | \( 1 + (0.382 - 0.923i)T \) | |
61 | \( 1 + (-0.773 - 0.634i)T \) | |
67 | \( 1 + (-0.980 - 0.195i)T \) | |
71 | \( 1 + (0.773 - 0.634i)T \) | |
73 | \( 1 + (0.290 - 0.956i)T \) | |
79 | \( 1 + (-0.995 + 0.0980i)T \) | |
83 | \( 1 + (0.555 + 0.831i)T \) | |
89 | \( 1 + (-0.471 - 0.881i)T \) | |
97 | \( 1 + (0.195 + 0.980i)T \) | |
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Imaginary part of the first few zeros on the critical line
−27.41135158325732176071228397263, −26.078806236527931663543435056726, −25.62901760367068835765176256185, −24.95165780418670401086004811052, −23.758701528727003948443773591843, −22.645087212172930801764543035320, −21.79845355279559303618698682460, −21.09868356345510952578465677263, −19.96911414824461302656451909433, −18.66772233000170992642096557304, −17.434884394385935381046414564047, −16.652217748291649082066659278464, −15.68263368411148859236295419923, −15.17441487510013182720274337043, −13.80089890049224582103516979089, −13.16409768185798782705501969095, −11.88644167935546520756720904613, −10.0428376320114034096175633037, −9.10828738919634324514520750731, −8.81293483221606614179695861991, −7.11571468414267007169943557485, −5.84852654084053738334663563542, −4.916771680043656327425158661167, −3.97186566192984706343160481224, −2.32925645514180319673120335203, 0.27758290109487275252878505005, 1.686632011331462706917521445045, 2.93832019417416860664832610571, 3.69218559840666334354404915939, 5.79880192819521124269155566231, 6.55664463985247067331852427135, 8.09716023767826439634989746374, 9.162661366984756977334889393242, 10.526856627683666273224146888533, 11.00387989635926644059569061335, 12.736615441607396540567652970226, 13.12355477981472056794254586701, 14.08237387894724553004237052852, 14.871089530173058313435456737708, 16.78368919887972059599467460894, 17.90773862047928063802463570246, 18.61701013809744104367981409341, 19.4312418488011033032254061479, 20.24355183047915097129923805755, 21.31317280325733803068394202922, 22.381813091112310146089414366780, 23.12563143884382700653748625706, 24.0447975397969382370255431715, 25.32977812788148654657422893852, 26.23335139382795763677863900728