Dirichlet series
L(s) = 1 | + (−0.964 − 0.264i)2-s + (0.944 − 0.328i)3-s + (0.860 + 0.509i)4-s + (−0.784 − 0.619i)5-s + (−0.997 + 0.0667i)6-s + (0.231 + 0.972i)7-s + (−0.695 − 0.718i)8-s + (0.784 − 0.619i)9-s + (0.593 + 0.805i)10-s + (0.860 − 0.509i)11-s + (0.979 + 0.199i)12-s + (−0.997 + 0.0667i)13-s + (0.0334 − 0.999i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (0.0334 + 0.999i)17-s + ⋯ |
L(s) = 1 | + (−0.964 − 0.264i)2-s + (0.944 − 0.328i)3-s + (0.860 + 0.509i)4-s + (−0.784 − 0.619i)5-s + (−0.997 + 0.0667i)6-s + (0.231 + 0.972i)7-s + (−0.695 − 0.718i)8-s + (0.784 − 0.619i)9-s + (0.593 + 0.805i)10-s + (0.860 − 0.509i)11-s + (0.979 + 0.199i)12-s + (−0.997 + 0.0667i)13-s + (0.0334 − 0.999i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (0.0334 + 0.999i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(283\) |
Sign: | $0.486 - 0.873i$ |
Analytic conductor: | \(30.4125\) |
Root analytic conductor: | \(30.4125\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{283} (128, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 283,\ (1:\ ),\ 0.486 - 0.873i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(1.387627809 - 0.8157344867i\) |
\(L(\frac12)\) | \(\approx\) | \(1.387627809 - 0.8157344867i\) |
\(L(1)\) | \(\approx\) | \(0.9342330571 - 0.2708012619i\) |
\(L(1)\) | \(\approx\) | \(0.9342330571 - 0.2708012619i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (-0.964 - 0.264i)T \) |
3 | \( 1 + (0.944 - 0.328i)T \) | |
5 | \( 1 + (-0.784 - 0.619i)T \) | |
7 | \( 1 + (0.231 + 0.972i)T \) | |
11 | \( 1 + (0.860 - 0.509i)T \) | |
13 | \( 1 + (-0.997 + 0.0667i)T \) | |
17 | \( 1 + (0.0334 + 0.999i)T \) | |
19 | \( 1 + (0.997 - 0.0667i)T \) | |
23 | \( 1 + (-0.420 - 0.907i)T \) | |
29 | \( 1 + (0.920 + 0.390i)T \) | |
31 | \( 1 + (-0.359 - 0.933i)T \) | |
37 | \( 1 + (0.645 - 0.763i)T \) | |
41 | \( 1 + (0.695 - 0.718i)T \) | |
43 | \( 1 + (0.166 + 0.986i)T \) | |
47 | \( 1 + (0.997 + 0.0667i)T \) | |
53 | \( 1 + (-0.480 + 0.876i)T \) | |
59 | \( 1 + (-0.892 + 0.451i)T \) | |
61 | \( 1 + (0.593 - 0.805i)T \) | |
67 | \( 1 + (0.979 - 0.199i)T \) | |
71 | \( 1 + (0.860 - 0.509i)T \) | |
73 | \( 1 + (0.359 - 0.933i)T \) | |
79 | \( 1 + (0.166 - 0.986i)T \) | |
83 | \( 1 + (-0.538 + 0.842i)T \) | |
89 | \( 1 + (-0.166 - 0.986i)T \) | |
97 | \( 1 + (-0.420 - 0.907i)T \) | |
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Imaginary part of the first few zeros on the critical line
−25.65352026034580330911457586974, −24.872774665858767831965779844354, −23.97006940880081095489667669675, −22.9216202997308880657663902961, −21.78961221033634407165681886427, −20.376281320360961650128104886721, −19.962171044333831825896298388306, −19.34839382879299561528492980303, −18.2657543952844298016318629691, −17.30435085818888065972213948799, −16.196619496281050070178456639947, −15.472109768116054636261729409981, −14.452811659029534026856510275980, −14.00116520382958551088854095477, −12.100795474626777798862080063258, −11.19788869736185256931146157342, −10.02574925782143158049196066260, −9.54084958499519569126164184261, −8.17977406231337132474197373484, −7.367712853310221864399971833094, −6.92569954302342220769403255261, −4.86097370220016738991185096223, −3.6345134729804328572549410619, −2.52079934781418347708207536082, −1.02713254855383012016065973969, 0.756664366351916705011363576575, 1.98450023893817249885566624757, 3.096201081303930949507259901673, 4.272843957105723764689945085592, 6.101494733584078798989110500566, 7.422610551963939305753037473431, 8.178308126665511030825002745854, 8.9499462193779526578549326853, 9.60705126944814581772453293341, 11.14901205565266942218000210451, 12.27284208087061231481956560761, 12.52119301560930140667443517594, 14.28304600639159092058874167799, 15.166860615143180270955579969787, 16.010030323509308429282071204841, 17.00793191913606484494126943040, 18.14695865398321207287206797936, 19.035354951197366741953969453038, 19.63890656886030541776292026944, 20.27478560011917444240637281335, 21.33451323661574198793034882011, 22.13331969428346316675922670719, 24.01537707970516186202086473760, 24.55291696298360578327614887275, 25.08061891884508381715375712952