Dirichlet series
L(s) = 1 | + (0.697 − 0.716i)2-s + (−0.364 − 0.931i)3-s + (−0.0265 − 0.999i)4-s + (−0.778 + 0.627i)5-s + (−0.921 − 0.388i)6-s + (−0.587 + 0.809i)7-s + (−0.734 − 0.678i)8-s + (−0.733 + 0.679i)9-s + (−0.0936 + 0.995i)10-s + (0.980 − 0.197i)11-s + (−0.921 + 0.389i)12-s + (0.209 + 0.977i)13-s + (0.169 + 0.985i)14-s + (0.868 + 0.495i)15-s + (−0.998 + 0.0531i)16-s + (0.191 − 0.981i)17-s + ⋯ |
L(s) = 1 | + (0.697 − 0.716i)2-s + (−0.364 − 0.931i)3-s + (−0.0265 − 0.999i)4-s + (−0.778 + 0.627i)5-s + (−0.921 − 0.388i)6-s + (−0.587 + 0.809i)7-s + (−0.734 − 0.678i)8-s + (−0.733 + 0.679i)9-s + (−0.0936 + 0.995i)10-s + (0.980 − 0.197i)11-s + (−0.921 + 0.389i)12-s + (0.209 + 0.977i)13-s + (0.169 + 0.985i)14-s + (0.868 + 0.495i)15-s + (−0.998 + 0.0531i)16-s + (0.191 − 0.981i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(4021\) |
Sign: | $0.740 - 0.672i$ |
Analytic conductor: | \(432.116\) |
Root analytic conductor: | \(432.116\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{4021} (30, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 4021,\ (1:\ ),\ 0.740 - 0.672i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(1.171258346 - 0.4524577791i\) |
\(L(\frac12)\) | \(\approx\) | \(1.171258346 - 0.4524577791i\) |
\(L(1)\) | \(\approx\) | \(0.8296182764 - 0.5183905781i\) |
\(L(1)\) | \(\approx\) | \(0.8296182764 - 0.5183905781i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 4021 | \( 1 \) |
good | 2 | \( 1 + (0.697 - 0.716i)T \) |
3 | \( 1 + (-0.364 - 0.931i)T \) | |
5 | \( 1 + (-0.778 + 0.627i)T \) | |
7 | \( 1 + (-0.587 + 0.809i)T \) | |
11 | \( 1 + (0.980 - 0.197i)T \) | |
13 | \( 1 + (0.209 + 0.977i)T \) | |
17 | \( 1 + (0.191 - 0.981i)T \) | |
19 | \( 1 + (0.829 + 0.558i)T \) | |
23 | \( 1 + (-0.763 - 0.645i)T \) | |
29 | \( 1 + (-0.873 + 0.486i)T \) | |
31 | \( 1 + (0.808 - 0.589i)T \) | |
37 | \( 1 + (-0.978 + 0.207i)T \) | |
41 | \( 1 + (-0.738 - 0.673i)T \) | |
43 | \( 1 + (0.00625 - 0.999i)T \) | |
47 | \( 1 + (-0.866 + 0.5i)T \) | |
53 | \( 1 + (-0.998 - 0.0624i)T \) | |
59 | \( 1 + (-0.611 + 0.791i)T \) | |
61 | \( 1 + (0.638 - 0.769i)T \) | |
67 | \( 1 + (-0.472 - 0.881i)T \) | |
71 | \( 1 + (-0.335 + 0.941i)T \) | |
73 | \( 1 + (-0.291 - 0.956i)T \) | |
79 | \( 1 + (0.600 + 0.799i)T \) | |
83 | \( 1 + (0.215 - 0.976i)T \) | |
89 | \( 1 + (0.197 + 0.980i)T \) | |
97 | \( 1 + (-0.821 - 0.569i)T \) | |
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Imaginary part of the first few zeros on the critical line
−17.92974112848847425808555379610, −17.30740846454675818531651736204, −16.920788201621549191575655273389, −16.13116762297222875174406581400, −15.79317203633675559448822396222, −15.115953718999081388863300383467, −14.51668152750272125350931249983, −13.62212453250664050219913476574, −12.98296538154190719429511214546, −12.21682102011219560381020162942, −11.65602455846799932784678551433, −10.97644867002084046023811106417, −9.980851481042977029004460072752, −9.378597979185671266031737301426, −8.48551858093188558529720458681, −7.90350735018958904894907249913, −7.05633796969638331210524802948, −6.274385073642124521620173674689, −5.5986472783955426785933663144, −4.77636184569839264625815093559, −4.20454111488508013020321301282, −3.42697668388716153922994339926, −3.27271001419299284522215801328, −1.36726322428488081676241523084, −0.28079980165281091757848429460, 0.492172895527900391400456950171, 1.56907862141966002789613790949, 2.24755716528810777235186718940, 3.15770670318842887091603784673, 3.65386479941318495077258319747, 4.67254212648562014214130759391, 5.539553472862434829185393263528, 6.35147414345893657655103698289, 6.68620700377563912301262839844, 7.49562165932145261649345381578, 8.5608249430022112530958695830, 9.27084092610266067047017041807, 10.07638976145575554504734855357, 11.03864923421190465454976340466, 11.67249011385627117563557362742, 12.012580222230272085369697274494, 12.39577556686678991546721398104, 13.48652529228253585068786448327, 14.0774981032274960538643056471, 14.4708003139197863181005980092, 15.44040730467443187411888293420, 16.121951814573379154718008541452, 16.7262350295989159302866979480, 17.96409028056569833111232937096, 18.55398737033210496693067993464