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: | $\chi_{193} (173, \cdot )$ |
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
−26.23335139382795763677863900728, −25.32977812788148654657422893852, −24.0447975397969382370255431715, −23.12563143884382700653748625706, −22.381813091112310146089414366780, −21.31317280325733803068394202922, −20.24355183047915097129923805755, −19.4312418488011033032254061479, −18.61701013809744104367981409341, −17.90773862047928063802463570246, −16.78368919887972059599467460894, −14.871089530173058313435456737708, −14.08237387894724553004237052852, −13.12355477981472056794254586701, −12.736615441607396540567652970226, −11.00387989635926644059569061335, −10.526856627683666273224146888533, −9.162661366984756977334889393242, −8.09716023767826439634989746374, −6.55664463985247067331852427135, −5.79880192819521124269155566231, −3.69218559840666334354404915939, −2.93832019417416860664832610571, −1.686632011331462706917521445045, −0.27758290109487275252878505005, 2.32925645514180319673120335203, 3.97186566192984706343160481224, 4.916771680043656327425158661167, 5.84852654084053738334663563542, 7.11571468414267007169943557485, 8.81293483221606614179695861991, 9.10828738919634324514520750731, 10.0428376320114034096175633037, 11.88644167935546520756720904613, 13.16409768185798782705501969095, 13.80089890049224582103516979089, 15.17441487510013182720274337043, 15.68263368411148859236295419923, 16.652217748291649082066659278464, 17.434884394385935381046414564047, 18.66772233000170992642096557304, 19.96911414824461302656451909433, 21.09868356345510952578465677263, 21.79845355279559303618698682460, 22.645087212172930801764543035320, 23.758701528727003948443773591843, 24.95165780418670401086004811052, 25.62901760367068835765176256185, 26.078806236527931663543435056726, 27.41135158325732176071228397263