Dirichlet series
L(s) = 1 | + (−0.831 − 0.555i)2-s + (0.923 − 0.382i)3-s + (0.382 + 0.923i)4-s + (0.956 + 0.290i)5-s + (−0.980 − 0.195i)6-s + (0.707 − 0.707i)7-s + (0.195 − 0.980i)8-s + (0.707 − 0.707i)9-s + (−0.634 − 0.773i)10-s + (−0.881 − 0.471i)11-s + (0.707 + 0.707i)12-s + (−0.773 − 0.634i)13-s + (−0.980 + 0.195i)14-s + (0.995 − 0.0980i)15-s + (−0.707 + 0.707i)16-s + (−0.956 + 0.290i)17-s + ⋯ |
L(s) = 1 | + (−0.831 − 0.555i)2-s + (0.923 − 0.382i)3-s + (0.382 + 0.923i)4-s + (0.956 + 0.290i)5-s + (−0.980 − 0.195i)6-s + (0.707 − 0.707i)7-s + (0.195 − 0.980i)8-s + (0.707 − 0.707i)9-s + (−0.634 − 0.773i)10-s + (−0.881 − 0.471i)11-s + (0.707 + 0.707i)12-s + (−0.773 − 0.634i)13-s + (−0.980 + 0.195i)14-s + (0.995 − 0.0980i)15-s + (−0.707 + 0.707i)16-s + (−0.956 + 0.290i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(193\) |
Sign: | $-0.461 - 0.887i$ |
Analytic conductor: | \(20.7407\) |
Root analytic conductor: | \(20.7407\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{193} (158, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 193,\ (1:\ ),\ -0.461 - 0.887i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.9836422782 - 1.620974234i\) |
\(L(\frac12)\) | \(\approx\) | \(0.9836422782 - 1.620974234i\) |
\(L(1)\) | \(\approx\) | \(1.007495341 - 0.6028300774i\) |
\(L(1)\) | \(\approx\) | \(1.007495341 - 0.6028300774i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 193 | \( 1 \) |
good | 2 | \( 1 + (-0.831 - 0.555i)T \) |
3 | \( 1 + (0.923 - 0.382i)T \) | |
5 | \( 1 + (0.956 + 0.290i)T \) | |
7 | \( 1 + (0.707 - 0.707i)T \) | |
11 | \( 1 + (-0.881 - 0.471i)T \) | |
13 | \( 1 + (-0.773 - 0.634i)T \) | |
17 | \( 1 + (-0.956 + 0.290i)T \) | |
19 | \( 1 + (0.290 - 0.956i)T \) | |
23 | \( 1 + (-0.831 - 0.555i)T \) | |
29 | \( 1 + (0.0980 - 0.995i)T \) | |
31 | \( 1 + (0.555 - 0.831i)T \) | |
37 | \( 1 + (0.0980 + 0.995i)T \) | |
41 | \( 1 + (0.881 + 0.471i)T \) | |
43 | \( 1 + (0.707 + 0.707i)T \) | |
47 | \( 1 + (-0.634 + 0.773i)T \) | |
53 | \( 1 + (-0.773 - 0.634i)T \) | |
59 | \( 1 + (0.923 - 0.382i)T \) | |
61 | \( 1 + (0.290 + 0.956i)T \) | |
67 | \( 1 + (0.555 - 0.831i)T \) | |
71 | \( 1 + (-0.290 + 0.956i)T \) | |
73 | \( 1 + (-0.0980 + 0.995i)T \) | |
79 | \( 1 + (0.471 - 0.881i)T \) | |
83 | \( 1 + (-0.195 - 0.980i)T \) | |
89 | \( 1 + (0.773 - 0.634i)T \) | |
97 | \( 1 + (-0.831 + 0.555i)T \) | |
show more | ||
show less |
Imaginary part of the first few zeros on the critical line
−26.882059763628157622858422897298, −26.16699379028544153025609862741, −25.2372873172840734438101709588, −24.69025254101329970613993046955, −23.854225001187602715287807256748, −22.1048897842628798510518717529, −21.13589271903536199943012390326, −20.438802062680674273259599203458, −19.3814530376565490206583813149, −18.22798611826604377295475910577, −17.69889576319928834441541703441, −16.3413995509414450557798248154, −15.54886882209890502653539865384, −14.4774431228733723192019960934, −13.90269829708816733040820368310, −12.398162875592194285826710976578, −10.769361813430441565577704665612, −9.80009915717342519572228579875, −9.07740657386640855873579262088, −8.17615832465547377772549265, −7.11704555699770306062664253480, −5.5577083984028894023590855249, −4.71902015265811565541374907044, −2.39834120328313444123401458409, −1.79225919631185230442506848734, 0.73599115478638206752600319946, 2.17120878947742185192138990392, 2.83878140384693991484726433853, 4.48035172123392693793580590009, 6.440490829691871431211575794641, 7.624578265573942829384408802962, 8.31206503535668233529453058919, 9.58074508458073249754053687914, 10.32130998173061371015336023153, 11.37610911079479557021011002694, 12.98311010719430164153765016629, 13.46782108973462455107793762474, 14.65331246105330798793169919454, 15.854783342974668923240166952271, 17.44149348696155806402209321785, 17.77459806412808960707812394912, 18.81197733231765069770396603935, 19.84073298389227532396938586383, 20.619387919780295867002601408329, 21.32748077168762953706098603719, 22.33504818595055700139328110528, 24.22931218678197412886695445876, 24.62538088221439288866518825979, 26.024086833404096359948588302559, 26.32087370787715906843845862368