Dirichlet series
L(s) = 1 | + (0.998 − 0.0506i)2-s + (0.874 + 0.485i)3-s + (0.994 − 0.101i)4-s + (−0.651 − 0.758i)5-s + (0.897 + 0.440i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (0.528 + 0.848i)9-s + (−0.688 − 0.724i)10-s + (0.988 + 0.151i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (−0.790 + 0.612i)14-s + (−0.201 − 0.979i)15-s + (0.979 − 0.201i)16-s + (0.994 − 0.101i)17-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0506i)2-s + (0.874 + 0.485i)3-s + (0.994 − 0.101i)4-s + (−0.651 − 0.758i)5-s + (0.897 + 0.440i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (0.528 + 0.848i)9-s + (−0.688 − 0.724i)10-s + (0.988 + 0.151i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (−0.790 + 0.612i)14-s + (−0.201 − 0.979i)15-s + (0.979 − 0.201i)16-s + (0.994 − 0.101i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(373\) |
Sign: | $0.762 + 0.646i$ |
Analytic conductor: | \(40.0844\) |
Root analytic conductor: | \(40.0844\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{373} (140, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 373,\ (1:\ ),\ 0.762 + 0.646i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(4.680069577 + 1.716662170i\) |
\(L(\frac12)\) | \(\approx\) | \(4.680069577 + 1.716662170i\) |
\(L(1)\) | \(\approx\) | \(2.454998522 + 0.4101422949i\) |
\(L(1)\) | \(\approx\) | \(2.454998522 + 0.4101422949i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 373 | \( 1 \) |
good | 2 | \( 1 + (0.998 - 0.0506i)T \) |
3 | \( 1 + (0.874 + 0.485i)T \) | |
5 | \( 1 + (-0.651 - 0.758i)T \) | |
7 | \( 1 + (-0.820 + 0.571i)T \) | |
11 | \( 1 + (0.988 + 0.151i)T \) | |
13 | \( 1 + (-0.151 + 0.988i)T \) | |
17 | \( 1 + (0.994 - 0.101i)T \) | |
19 | \( 1 + (0.201 - 0.979i)T \) | |
23 | \( 1 + (0.485 + 0.874i)T \) | |
29 | \( 1 + (-0.250 - 0.968i)T \) | |
31 | \( 1 + (-0.918 + 0.394i)T \) | |
37 | \( 1 + (0.954 - 0.299i)T \) | |
41 | \( 1 + (-0.440 + 0.897i)T \) | |
43 | \( 1 + (0.101 + 0.994i)T \) | |
47 | \( 1 + (0.790 + 0.612i)T \) | |
53 | \( 1 + (0.848 + 0.528i)T \) | |
59 | \( 1 + (0.250 + 0.968i)T \) | |
61 | \( 1 + (0.485 - 0.874i)T \) | |
67 | \( 1 + (-0.651 - 0.758i)T \) | |
71 | \( 1 + (0.994 + 0.101i)T \) | |
73 | \( 1 + (0.528 - 0.848i)T \) | |
79 | \( 1 + (-0.724 + 0.688i)T \) | |
83 | \( 1 + (0.528 - 0.848i)T \) | |
89 | \( 1 - T \) | |
97 | \( 1 + (-0.101 - 0.994i)T \) | |
show more | ||
show less |
Imaginary part of the first few zeros on the critical line
−24.175216490542011393479023009775, −23.34879621683163781676434722747, −22.6410332106588553090711237585, −21.954353026505185577573653160763, −20.51851719782946413723597756054, −20.09254963531709974357642770108, −19.22091280937862230949545083943, −18.54123281827996535738432777497, −16.93403831620541960628716484827, −16.08280142631800860432926666575, −14.90708682613874857320280051417, −14.564293550037776672707840774927, −13.65212882613578565694331061474, −12.62157865751085619666098268497, −12.07914151382796056966043434593, −10.74112620956789904279967396438, −9.89265112774899916336839716447, −8.34111711457822013200537751662, −7.33171281041651543113957316150, −6.82283798313517570912135593830, −5.71760444293937524842677006273, −3.81726157238450676471636131285, −3.58562583723195699283380053104, −2.54559918758439361873003675921, −0.976193213118569894226191600012, 1.40719644466162891869291675141, 2.760847416773699538730627682457, 3.71547062396296043500511834352, 4.448056222551536780045015004327, 5.51829281375893881845196799890, 6.85807957259167492473653906275, 7.78921382052440484516209691528, 9.16274848816547908228063177628, 9.59212382027959268848989513680, 11.23888088745529878302324297138, 12.008143484463187511110433905975, 12.91283074563620602685447370811, 13.73441806887597772903795419029, 14.7612795003657953372579608929, 15.40545665926926238473930103028, 16.30101832934680249276002278921, 16.80350976334259815316599597092, 18.90917451060699175558540864650, 19.61169365981171665135138031353, 20.02889536033665789988633359381, 21.2260209680391688626591183578, 21.68847652255284524363862306620, 22.67929675313248087430161366195, 23.60523411787611218501560997771, 24.553555538068306395220222600294