Dirichlet series
L(s) = 1 | + (0.0883 − 0.996i)2-s + (−0.186 + 0.982i)3-s + (−0.984 − 0.176i)4-s + (0.633 − 0.773i)5-s + (0.962 + 0.273i)6-s + (0.616 + 0.787i)7-s + (−0.262 + 0.964i)8-s + (−0.930 − 0.367i)9-s + (−0.714 − 0.699i)10-s + (0.752 + 0.658i)11-s + (0.356 − 0.934i)12-s + (−0.650 − 0.759i)13-s + (0.839 − 0.544i)14-s + (0.641 + 0.766i)15-s + (0.937 + 0.346i)16-s + (−0.801 + 0.598i)17-s + ⋯ |
L(s) = 1 | + (0.0883 − 0.996i)2-s + (−0.186 + 0.982i)3-s + (−0.984 − 0.176i)4-s + (0.633 − 0.773i)5-s + (0.962 + 0.273i)6-s + (0.616 + 0.787i)7-s + (−0.262 + 0.964i)8-s + (−0.930 − 0.367i)9-s + (−0.714 − 0.699i)10-s + (0.752 + 0.658i)11-s + (0.356 − 0.934i)12-s + (−0.650 − 0.759i)13-s + (0.839 − 0.544i)14-s + (0.641 + 0.766i)15-s + (0.937 + 0.346i)16-s + (−0.801 + 0.598i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(569\) |
Sign: | $-0.249 + 0.968i$ |
Analytic conductor: | \(61.1475\) |
Root analytic conductor: | \(61.1475\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{569} (363, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 569,\ (1:\ ),\ -0.249 + 0.968i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.5785146887 + 0.7467740296i\) |
\(L(\frac12)\) | \(\approx\) | \(0.5785146887 + 0.7467740296i\) |
\(L(1)\) | \(\approx\) | \(0.9451030694 - 0.07729121906i\) |
\(L(1)\) | \(\approx\) | \(0.9451030694 - 0.07729121906i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 569 | \( 1 \) |
good | 2 | \( 1 + (0.0883 - 0.996i)T \) |
3 | \( 1 + (-0.186 + 0.982i)T \) | |
5 | \( 1 + (0.633 - 0.773i)T \) | |
7 | \( 1 + (0.616 + 0.787i)T \) | |
11 | \( 1 + (0.752 + 0.658i)T \) | |
13 | \( 1 + (-0.650 - 0.759i)T \) | |
17 | \( 1 + (-0.801 + 0.598i)T \) | |
19 | \( 1 + (0.820 + 0.571i)T \) | |
23 | \( 1 + (-0.457 + 0.888i)T \) | |
29 | \( 1 + (-0.143 - 0.989i)T \) | |
31 | \( 1 + (-0.251 + 0.967i)T \) | |
37 | \( 1 + (-0.121 + 0.992i)T \) | |
41 | \( 1 + (0.952 - 0.304i)T \) | |
43 | \( 1 + (-0.996 + 0.0883i)T \) | |
47 | \( 1 + (-0.948 - 0.315i)T \) | |
53 | \( 1 + (0.273 - 0.962i)T \) | |
59 | \( 1 + (0.315 - 0.948i)T \) | |
61 | \( 1 + (-0.467 + 0.883i)T \) | |
67 | \( 1 + (-0.801 - 0.598i)T \) | |
71 | \( 1 + (0.262 + 0.964i)T \) | |
73 | \( 1 + (0.571 - 0.820i)T \) | |
79 | \( 1 + (-0.850 + 0.525i)T \) | |
83 | \( 1 + (-0.515 - 0.856i)T \) | |
89 | \( 1 + (-0.967 + 0.251i)T \) | |
97 | \( 1 + (-0.833 - 0.553i)T \) | |
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Imaginary part of the first few zeros on the critical line
−22.855517696205319693726786513105, −22.28661487056135735162713547202, −21.54297604630707951265457039844, −20.10402766032425137890347745501, −19.22935081538760912074694159705, −18.18831014790816620025365667466, −17.91119425653198175763389388038, −16.9267786997106178433986106010, −16.43071483528734726506405548630, −14.89090601052577089438849092129, −14.05244575343985492191984577681, −13.93427495385919461028209252593, −12.91842884647752028701380705090, −11.66090023382016518964094473932, −10.92048923943124646921145040752, −9.578293692601544825685417680452, −8.72442105061462216206826929425, −7.52422524574015995439384278748, −6.98941501631910042397118105887, −6.32548569611748080933279548433, −5.31463599477500516027871335239, −4.214245069803465426941312334755, −2.81583946913654034846484515064, −1.48982833437849618712050798764, −0.23204204251557268207692597669, 1.37741058132587585281325734337, 2.30471341969945272461642796479, 3.56746447176399182318186735684, 4.6328093532122613074622258306, 5.21364375984112371921675062807, 6.02846498283854384120376196029, 8.09978596943593999118136016069, 8.88065566775783348734541260590, 9.697380179450539462863603381925, 10.161140923259771583839044415806, 11.451200040314255670693800265235, 11.99353929735230440870042660671, 12.84539605193291978147803025653, 13.976437538344558876493470363036, 14.77982288586265466144249165536, 15.54586114946376422714254336163, 16.83727495830292912598411050578, 17.64143873211241273327455814365, 17.95827664509635809324373899713, 19.57396002749374388542086249371, 20.12355373739557015350883349696, 20.8940864596035331218173363974, 21.57262742454099205556965620220, 22.17548641018607081607588739577, 22.833768045020635682371873466895