Dirichlet series
L(s) = 1 | + (−0.980 + 0.195i)2-s + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)4-s + (0.995 − 0.0980i)5-s + (0.555 + 0.831i)6-s + (−0.707 + 0.707i)7-s + (−0.831 + 0.555i)8-s + (−0.707 + 0.707i)9-s + (−0.956 + 0.290i)10-s + (0.634 + 0.773i)11-s + (−0.707 − 0.707i)12-s + (0.290 − 0.956i)13-s + (0.555 − 0.831i)14-s + (−0.471 − 0.881i)15-s + (0.707 − 0.707i)16-s + (−0.995 − 0.0980i)17-s + ⋯ |
L(s) = 1 | + (−0.980 + 0.195i)2-s + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)4-s + (0.995 − 0.0980i)5-s + (0.555 + 0.831i)6-s + (−0.707 + 0.707i)7-s + (−0.831 + 0.555i)8-s + (−0.707 + 0.707i)9-s + (−0.956 + 0.290i)10-s + (0.634 + 0.773i)11-s + (−0.707 − 0.707i)12-s + (0.290 − 0.956i)13-s + (0.555 − 0.831i)14-s + (−0.471 − 0.881i)15-s + (0.707 − 0.707i)16-s + (−0.995 − 0.0980i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(193\) |
Sign: | $-0.770 - 0.637i$ |
Analytic conductor: | \(20.7407\) |
Root analytic conductor: | \(20.7407\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{193} (105, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 193,\ (1:\ ),\ -0.770 - 0.637i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.2033822511 - 0.5645485137i\) |
\(L(\frac12)\) | \(\approx\) | \(0.2033822511 - 0.5645485137i\) |
\(L(1)\) | \(\approx\) | \(0.5867373320 - 0.1789686842i\) |
\(L(1)\) | \(\approx\) | \(0.5867373320 - 0.1789686842i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 193 | \( 1 \) |
good | 2 | \( 1 + (-0.980 + 0.195i)T \) |
3 | \( 1 + (-0.382 - 0.923i)T \) | |
5 | \( 1 + (0.995 - 0.0980i)T \) | |
7 | \( 1 + (-0.707 + 0.707i)T \) | |
11 | \( 1 + (0.634 + 0.773i)T \) | |
13 | \( 1 + (0.290 - 0.956i)T \) | |
17 | \( 1 + (-0.995 - 0.0980i)T \) | |
19 | \( 1 + (-0.0980 - 0.995i)T \) | |
23 | \( 1 + (-0.980 + 0.195i)T \) | |
29 | \( 1 + (0.881 + 0.471i)T \) | |
31 | \( 1 + (-0.195 - 0.980i)T \) | |
37 | \( 1 + (0.881 - 0.471i)T \) | |
41 | \( 1 + (-0.634 - 0.773i)T \) | |
43 | \( 1 + (-0.707 - 0.707i)T \) | |
47 | \( 1 + (-0.956 - 0.290i)T \) | |
53 | \( 1 + (0.290 - 0.956i)T \) | |
59 | \( 1 + (-0.382 - 0.923i)T \) | |
61 | \( 1 + (-0.0980 + 0.995i)T \) | |
67 | \( 1 + (-0.195 - 0.980i)T \) | |
71 | \( 1 + (0.0980 + 0.995i)T \) | |
73 | \( 1 + (-0.881 - 0.471i)T \) | |
79 | \( 1 + (-0.773 + 0.634i)T \) | |
83 | \( 1 + (0.831 + 0.555i)T \) | |
89 | \( 1 + (-0.290 - 0.956i)T \) | |
97 | \( 1 + (-0.980 - 0.195i)T \) | |
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Imaginary part of the first few zeros on the critical line
−26.915376901899033353065801910578, −26.50973082672585445528052402547, −25.6510118872014876173628340068, −24.644311613644412343095226206882, −23.31136460431382598945384709477, −21.96386214217800054461775788774, −21.53083846871614517554345062881, −20.436310169556467247728286488872, −19.59690047173385157829259481199, −18.367503305339702004642223238, −17.38770182718181818186567993681, −16.53831390782570945900427324359, −16.13780495296486962345708211559, −14.59356869744884964196905768325, −13.525419057554691492378594988786, −11.99791926561601151187796327415, −10.95315072779265107229321013639, −10.12126886755636932594437373727, −9.421242784547125610742111596036, −8.47736733842600905897162956375, −6.512475832916919539302821591908, −6.19943619697577511034732440757, −4.221317461787987095540504876487, −3.03184544256131687156642200175, −1.36420831254100622073643655752, 0.3018492107949454590961095704, 1.794791282265032338497181997640, 2.65069611504646051670567548281, 5.34288642486381614973689922166, 6.30316227383307201349290762420, 6.93866838126156874236086363493, 8.38766242264321887703539122978, 9.30629526523327682162425157401, 10.3048277457741289562698457720, 11.53662051912043128404140512747, 12.5797817069488823915063196555, 13.48105768368260913558461408098, 14.90268402064492256107239318700, 15.992077824273923675213680233539, 17.15529735515429712677856510503, 17.853773570410740694067378039254, 18.36436751711261648799851417349, 19.6595260480361720694966147018, 20.20405036809422659680886696001, 21.81774836827095639621937286205, 22.60970545777688210789327014851, 23.9738725985243500467938183119, 24.8726935092598693037417040610, 25.428616694617820304143244842845, 26.05810120580421225732773830419