Dirichlet series
L(s) = 1 | + (−0.994 + 0.101i)2-s + (−0.994 − 0.101i)3-s + (0.979 − 0.201i)4-s + (0.688 + 0.724i)5-s + 6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (0.979 + 0.201i)9-s + (−0.758 − 0.651i)10-s + (0.0506 + 0.998i)11-s + (−0.994 + 0.101i)12-s + (0.979 + 0.201i)13-s + (−0.250 + 0.968i)14-s + (−0.612 − 0.790i)15-s + (0.918 − 0.394i)16-s + (0.758 + 0.651i)17-s + ⋯ |
L(s) = 1 | + (−0.994 + 0.101i)2-s + (−0.994 − 0.101i)3-s + (0.979 − 0.201i)4-s + (0.688 + 0.724i)5-s + 6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (0.979 + 0.201i)9-s + (−0.758 − 0.651i)10-s + (0.0506 + 0.998i)11-s + (−0.994 + 0.101i)12-s + (0.979 + 0.201i)13-s + (−0.250 + 0.968i)14-s + (−0.612 − 0.790i)15-s + (0.918 − 0.394i)16-s + (0.758 + 0.651i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(311\) |
Sign: | $0.371 + 0.928i$ |
Analytic conductor: | \(33.4215\) |
Root analytic conductor: | \(33.4215\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{311} (206, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 311,\ (1:\ ),\ 0.371 + 0.928i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.9048540990 + 0.6122517454i\) |
\(L(\frac12)\) | \(\approx\) | \(0.9048540990 + 0.6122517454i\) |
\(L(1)\) | \(\approx\) | \(0.6856489059 + 0.1475121719i\) |
\(L(1)\) | \(\approx\) | \(0.6856489059 + 0.1475121719i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (-0.994 + 0.101i)T \) |
3 | \( 1 + (-0.994 - 0.101i)T \) | |
5 | \( 1 + (0.688 + 0.724i)T \) | |
7 | \( 1 + (0.347 - 0.937i)T \) | |
11 | \( 1 + (0.0506 + 0.998i)T \) | |
13 | \( 1 + (0.979 + 0.201i)T \) | |
17 | \( 1 + (0.758 + 0.651i)T \) | |
19 | \( 1 + (-0.688 + 0.724i)T \) | |
23 | \( 1 + (0.954 - 0.299i)T \) | |
29 | \( 1 + (0.250 + 0.968i)T \) | |
31 | \( 1 + (0.758 - 0.651i)T \) | |
37 | \( 1 + (-0.347 - 0.937i)T \) | |
41 | \( 1 + (-0.528 - 0.848i)T \) | |
43 | \( 1 + (-0.347 + 0.937i)T \) | |
47 | \( 1 + (-0.250 - 0.968i)T \) | |
53 | \( 1 + (0.347 - 0.937i)T \) | |
59 | \( 1 + (-0.347 + 0.937i)T \) | |
61 | \( 1 + (-0.688 + 0.724i)T \) | |
67 | \( 1 + (0.528 - 0.848i)T \) | |
71 | \( 1 + (0.874 + 0.485i)T \) | |
73 | \( 1 + (0.820 - 0.571i)T \) | |
79 | \( 1 + (0.528 + 0.848i)T \) | |
83 | \( 1 + (-0.612 + 0.790i)T \) | |
89 | \( 1 + (0.347 + 0.937i)T \) | |
97 | \( 1 + (0.874 + 0.485i)T \) | |
show more | ||
show less |
Imaginary part of the first few zeros on the critical line
−24.91106051009643791919274780901, −24.21566226567802179651194333637, −23.21634465242036312765870901892, −21.69843909814459726265376864079, −21.346930352542665044630803294472, −20.559036090546870330825827817451, −19.01567526500630559624050198607, −18.49155832269946370191447901227, −17.507773576980350426264489002317, −16.93495765782660380209450065431, −15.991816030368337128985752560949, −15.352574281756787740414305251751, −13.62596227394073504074062019695, −12.53465822848824997716746600767, −11.647139875211704991151913242704, −10.93614319939657073018527048395, −9.843146428155982696323610133484, −8.91802214641144724625960682139, −8.16542158855448447263649646943, −6.543171858925352657045121368603, −5.85582449783903192534817642444, −4.919153466983595250563777648266, −3.01639448038303343184090965470, −1.51876132597866513190399591966, −0.63114450444946619071710485211, 1.09593720648970727789014245202, 1.938009513634569521271537607630, 3.763154165942292378462161115006, 5.32560235758411967417514665119, 6.467299506478768289034470391703, 6.983335360341502430044435116404, 8.072701236674001274022481375051, 9.56629578453340206357505422628, 10.579509966184468261213047190909, 10.70360787038104523469159910426, 11.97812704769027593636278194901, 13.08943175785547970379776947453, 14.42660706645037872627442618211, 15.301813542701793591253474316206, 16.65212648430057847418612890541, 17.05841909712155972094826641508, 17.960957028822274325818368308205, 18.5141239011620962236647306107, 19.53045255847156906337174496091, 20.93606970159771826839552501498, 21.25466444023285590590781136584, 22.87770659183374823101488856310, 23.26009992421294527647227954172, 24.36157561930511649028755696445, 25.45712741730670603527655096991