Dirichlet series
L(s) = 1 | + (−0.195 + 0.980i)2-s + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)4-s + (0.773 − 0.634i)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 + (0.471 + 0.881i)10-s + (0.995 − 0.0980i)11-s + (−0.707 + 0.707i)12-s + (−0.881 − 0.471i)13-s + (0.831 − 0.555i)14-s + (−0.290 − 0.956i)15-s + (0.707 + 0.707i)16-s + (−0.773 − 0.634i)17-s + ⋯ |
L(s) = 1 | + (−0.195 + 0.980i)2-s + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)4-s + (0.773 − 0.634i)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 + (0.471 + 0.881i)10-s + (0.995 − 0.0980i)11-s + (−0.707 + 0.707i)12-s + (−0.881 − 0.471i)13-s + (0.831 − 0.555i)14-s + (−0.290 − 0.956i)15-s + (0.707 + 0.707i)16-s + (−0.773 − 0.634i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(193\) |
Sign: | $-0.653 - 0.756i$ |
Analytic conductor: | \(20.7407\) |
Root analytic conductor: | \(20.7407\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{193} (160, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 193,\ (1:\ ),\ -0.653 - 0.756i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.4539526443 - 0.9915560679i\) |
\(L(\frac12)\) | \(\approx\) | \(0.4539526443 - 0.9915560679i\) |
\(L(1)\) | \(\approx\) | \(0.9037505052 - 0.2167164627i\) |
\(L(1)\) | \(\approx\) | \(0.9037505052 - 0.2167164627i\) |
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 | \( 1 + (0.773 - 0.634i)T \) | |
7 | \( 1 + (-0.707 - 0.707i)T \) | |
11 | \( 1 + (0.995 - 0.0980i)T \) | |
13 | \( 1 + (-0.881 - 0.471i)T \) | |
17 | \( 1 + (-0.773 - 0.634i)T \) | |
19 | \( 1 + (0.634 + 0.773i)T \) | |
23 | \( 1 + (-0.195 + 0.980i)T \) | |
29 | \( 1 + (-0.956 - 0.290i)T \) | |
31 | \( 1 + (0.980 + 0.195i)T \) | |
37 | \( 1 + (-0.956 + 0.290i)T \) | |
41 | \( 1 + (-0.995 + 0.0980i)T \) | |
43 | \( 1 + (-0.707 + 0.707i)T \) | |
47 | \( 1 + (0.471 - 0.881i)T \) | |
53 | \( 1 + (-0.881 - 0.471i)T \) | |
59 | \( 1 + (0.382 - 0.923i)T \) | |
61 | \( 1 + (0.634 - 0.773i)T \) | |
67 | \( 1 + (0.980 + 0.195i)T \) | |
71 | \( 1 + (-0.634 - 0.773i)T \) | |
73 | \( 1 + (0.956 + 0.290i)T \) | |
79 | \( 1 + (-0.0980 - 0.995i)T \) | |
83 | \( 1 + (-0.555 - 0.831i)T \) | |
89 | \( 1 + (0.881 - 0.471i)T \) | |
97 | \( 1 + (-0.195 - 0.980i)T \) | |
show more | ||
show less |
Imaginary part of the first few zeros on the critical line
−27.07226068145833695821614620032, −26.36703245883222617717784009335, −25.69129675829949731543985084271, −24.53556416881634493111252940113, −22.46755878724870101455114787912, −22.23543963845352827828834303151, −21.62387170023965593012738522661, −20.47267888075086833613902306652, −19.53920554243228881385646098663, −18.82481278208798606197016067284, −17.52425830742024556477392450801, −16.760915722553739799271452776210, −15.27675394468493643669657266504, −14.35986712435015109439994141402, −13.48030980963872153004672817053, −12.18985855509177528947353029575, −11.12304573064605990960423546878, −10.06637538112580472976774156947, −9.39189781931645393981821546727, −8.689502744604720414836154504126, −6.77589728103109603453584049252, −5.32914778702761055526393085019, −4.06532958185423639622233643106, −2.895568464979060990059147025620, −2.06556624581338804680801795854, 0.3764996572907518088108919320, 1.605073533273761395594055812583, 3.533745771424394097377198737078, 5.124470413546158854814558044293, 6.302305232570523599566780352008, 7.056444778526181448255918425477, 8.1442245924282605417024249489, 9.319979107039724529013564807382, 9.9008057872847307169648687944, 11.974717541888308403313943056203, 13.1316703970496371022984977885, 13.70561714398951782018652720928, 14.52808277712098938569913448810, 15.90127883902868931662834650400, 17.14501245815985441774935581817, 17.376791656848574457149102498844, 18.63321568773363200395507114511, 19.672142371676825060365001744187, 20.35306196550486250233716985764, 22.09606499224561535416413697375, 22.84975374630996171561759287520, 23.978514837378943436443201892854, 24.830062780741434512792596573497, 25.15662773131633709963436976787, 26.25257201293400311112340890720