Dirichlet series
L(s) = 1 | + (0.528 − 0.848i)2-s + (0.528 + 0.848i)3-s + (−0.440 − 0.897i)4-s + (−0.250 + 0.968i)5-s + 6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (−0.440 + 0.897i)9-s + (0.688 + 0.724i)10-s + (0.874 − 0.485i)11-s + (0.528 − 0.848i)12-s + (−0.440 + 0.897i)13-s + (0.820 − 0.571i)14-s + (−0.954 + 0.299i)15-s + (−0.612 + 0.790i)16-s + (−0.688 − 0.724i)17-s + ⋯ |
L(s) = 1 | + (0.528 − 0.848i)2-s + (0.528 + 0.848i)3-s + (−0.440 − 0.897i)4-s + (−0.250 + 0.968i)5-s + 6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (−0.440 + 0.897i)9-s + (0.688 + 0.724i)10-s + (0.874 − 0.485i)11-s + (0.528 − 0.848i)12-s + (−0.440 + 0.897i)13-s + (0.820 − 0.571i)14-s + (−0.954 + 0.299i)15-s + (−0.612 + 0.790i)16-s + (−0.688 − 0.724i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(311\) |
Sign: | $0.111 + 0.993i$ |
Analytic conductor: | \(33.4215\) |
Root analytic conductor: | \(33.4215\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{311} (293, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 311,\ (1:\ ),\ 0.111 + 0.993i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(1.814562222 + 1.623139459i\) |
\(L(\frac12)\) | \(\approx\) | \(1.814562222 + 1.623139459i\) |
\(L(1)\) | \(\approx\) | \(1.502378721 + 0.2159610677i\) |
\(L(1)\) | \(\approx\) | \(1.502378721 + 0.2159610677i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (0.528 - 0.848i)T \) |
3 | \( 1 + (0.528 + 0.848i)T \) | |
5 | \( 1 + (-0.250 + 0.968i)T \) | |
7 | \( 1 + (0.918 + 0.394i)T \) | |
11 | \( 1 + (0.874 - 0.485i)T \) | |
13 | \( 1 + (-0.440 + 0.897i)T \) | |
17 | \( 1 + (-0.688 - 0.724i)T \) | |
19 | \( 1 + (0.250 + 0.968i)T \) | |
23 | \( 1 + (0.994 + 0.101i)T \) | |
29 | \( 1 + (-0.820 - 0.571i)T \) | |
31 | \( 1 + (-0.688 + 0.724i)T \) | |
37 | \( 1 + (-0.918 + 0.394i)T \) | |
41 | \( 1 + (0.758 + 0.651i)T \) | |
43 | \( 1 + (-0.918 - 0.394i)T \) | |
47 | \( 1 + (0.820 + 0.571i)T \) | |
53 | \( 1 + (0.918 + 0.394i)T \) | |
59 | \( 1 + (-0.918 - 0.394i)T \) | |
61 | \( 1 + (0.250 + 0.968i)T \) | |
67 | \( 1 + (-0.758 + 0.651i)T \) | |
71 | \( 1 + (-0.347 + 0.937i)T \) | |
73 | \( 1 + (0.979 + 0.201i)T \) | |
79 | \( 1 + (-0.758 - 0.651i)T \) | |
83 | \( 1 + (-0.954 - 0.299i)T \) | |
89 | \( 1 + (0.918 - 0.394i)T \) | |
97 | \( 1 + (-0.347 + 0.937i)T \) | |
show more | ||
show less |
Imaginary part of the first few zeros on the critical line
−24.472444408227248934936348403790, −24.29519274708312144382729785967, −23.36515926913476395901631157124, −22.39453614646495373592358439498, −21.17160423249795399273272848032, −20.25419229714586555111807275272, −19.69793857292719543414407852979, −18.16544942010678457537897130364, −17.33290307628592189676759524128, −16.928115936374096531483014510314, −15.29875922000161327714792938790, −14.8675813840692841741190122107, −13.76100150141371810244325826127, −12.96053158885630516415455363672, −12.304695672605606785476322876493, −11.25084663623300576664944726847, −9.19856254515774479253881300932, −8.610112264643099446530227437252, −7.59755580445603947319384520918, −6.99264021207980083041530901547, −5.57917818983803271255739611758, −4.60218467496045686789290768238, −3.573186446434399476930019761898, −1.9342403308134976722641325412, −0.54511246795789153135133307230, 1.73336100787848026469550272276, 2.77250880787584977947531650885, 3.7750583501833145124741732070, 4.63224084679118904730808104072, 5.76624136980669560326355283894, 7.19313108823771141510111380046, 8.693188663252170253026021985, 9.41962506441641744163984076888, 10.5456203296515843124003078714, 11.36425807538214775193851506413, 11.86414593301300296351256889855, 13.60816606668047272623613764522, 14.398429272013038598464271539081, 14.75522616898718508412601608154, 15.76108490859953025264680366534, 17.10774146113113773966643016353, 18.44398741619295923283326639075, 19.08104794619720501192801560245, 19.97023013925481353186495003855, 20.919617202802256176086907993297, 21.6596895065324617038145927992, 22.25165280537161460769717612554, 23.06331822133797913401918650611, 24.333993058932923970298844621702, 25.084659767122532365672221808756