Dirichlet series
L(s) = 1 | + (−0.954 + 0.299i)2-s + (−0.954 − 0.299i)3-s + (0.820 − 0.571i)4-s + (−0.758 + 0.651i)5-s + 6-s + (−0.874 + 0.485i)7-s + (−0.612 + 0.790i)8-s + (0.820 + 0.571i)9-s + (0.528 − 0.848i)10-s + (−0.151 − 0.988i)11-s + (−0.954 + 0.299i)12-s + (0.820 + 0.571i)13-s + (0.688 − 0.724i)14-s + (0.918 − 0.394i)15-s + (0.347 − 0.937i)16-s + (−0.528 + 0.848i)17-s + ⋯ |
L(s) = 1 | + (−0.954 + 0.299i)2-s + (−0.954 − 0.299i)3-s + (0.820 − 0.571i)4-s + (−0.758 + 0.651i)5-s + 6-s + (−0.874 + 0.485i)7-s + (−0.612 + 0.790i)8-s + (0.820 + 0.571i)9-s + (0.528 − 0.848i)10-s + (−0.151 − 0.988i)11-s + (−0.954 + 0.299i)12-s + (0.820 + 0.571i)13-s + (0.688 − 0.724i)14-s + (0.918 − 0.394i)15-s + (0.347 − 0.937i)16-s + (−0.528 + 0.848i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(311\) |
Sign: | $-0.606 + 0.795i$ |
Analytic conductor: | \(33.4215\) |
Root analytic conductor: | \(33.4215\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{311} (228, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 311,\ (1:\ ),\ -0.606 + 0.795i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.1714132067 + 0.3463504552i\) |
\(L(\frac12)\) | \(\approx\) | \(0.1714132067 + 0.3463504552i\) |
\(L(1)\) | \(\approx\) | \(0.4132084702 + 0.1034908339i\) |
\(L(1)\) | \(\approx\) | \(0.4132084702 + 0.1034908339i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (-0.954 + 0.299i)T \) |
3 | \( 1 + (-0.954 - 0.299i)T \) | |
5 | \( 1 + (-0.758 + 0.651i)T \) | |
7 | \( 1 + (-0.874 + 0.485i)T \) | |
11 | \( 1 + (-0.151 - 0.988i)T \) | |
13 | \( 1 + (0.820 + 0.571i)T \) | |
17 | \( 1 + (-0.528 + 0.848i)T \) | |
19 | \( 1 + (0.758 + 0.651i)T \) | |
23 | \( 1 + (0.612 - 0.790i)T \) | |
29 | \( 1 + (-0.688 - 0.724i)T \) | |
31 | \( 1 + (-0.528 - 0.848i)T \) | |
37 | \( 1 + (0.874 + 0.485i)T \) | |
41 | \( 1 + (0.994 - 0.101i)T \) | |
43 | \( 1 + (0.874 - 0.485i)T \) | |
47 | \( 1 + (0.688 + 0.724i)T \) | |
53 | \( 1 + (-0.874 + 0.485i)T \) | |
59 | \( 1 + (0.874 - 0.485i)T \) | |
61 | \( 1 + (0.758 + 0.651i)T \) | |
67 | \( 1 + (-0.994 - 0.101i)T \) | |
71 | \( 1 + (0.0506 + 0.998i)T \) | |
73 | \( 1 + (-0.250 - 0.968i)T \) | |
79 | \( 1 + (-0.994 + 0.101i)T \) | |
83 | \( 1 + (0.918 + 0.394i)T \) | |
89 | \( 1 + (-0.874 - 0.485i)T \) | |
97 | \( 1 + (0.0506 + 0.998i)T \) | |
show more | ||
show less |
Imaginary part of the first few zeros on the critical line
−24.82460236904407142816169039663, −23.6702908034825018825467952962, −22.96444780443214071100868730774, −22.084533935743456072156561829480, −20.75309004238098356182574094953, −20.22293794619588139731248330707, −19.37496200079123452075258908341, −18.137969306911378858779994826664, −17.58531586139576313611624257463, −16.43868133361082017164593484320, −15.97767323174415897253563229951, −15.30825399328722084300644757309, −13.07279499450020040247163377441, −12.571737163823900310450562180074, −11.44043225410318642141849625549, −10.8384230549621013328077135754, −9.65481037098545496575000339736, −9.06182082449756977348752430705, −7.45575774114904385636589839958, −6.98203683635694337268866465893, −5.519888259332949944180836370673, −4.23219149468859002695082961475, −3.17532124220374282851038683536, −1.1889315124519283425387859323, −0.27852711435939422442588696898, 0.873028989268912590299184005762, 2.50506600486463483491264716517, 3.92385975260159167762231502982, 5.87112878034377179107920079570, 6.23861187720922781682206645393, 7.297451072841415569405832364917, 8.28242507850255468791083047891, 9.42382655135846447591474619474, 10.674542625115862183249622548343, 11.18326212701151657170908149347, 12.06099711225947638806994904969, 13.24773613836857149093688891803, 14.69638600198447481292381436800, 15.858107095620966361365267349601, 16.180253954346625095108953212094, 17.14286501139143453691518156771, 18.42633075242402357725839515714, 18.800095392179036613937761106620, 19.35927925379338006869857883657, 20.75013091780063074027135199444, 22.01594328715890632827445379885, 22.749100622819712387827518744116, 23.75112259262925910261611687306, 24.31373988272568649803147718385, 25.44656989999947356542405476456