Dirichlet series
L(s) = 1 | + (0.151 + 0.988i)2-s + (0.151 − 0.988i)3-s + (−0.954 + 0.299i)4-s + (0.347 + 0.937i)5-s + 6-s + (−0.250 − 0.968i)7-s + (−0.440 − 0.897i)8-s + (−0.954 − 0.299i)9-s + (−0.874 + 0.485i)10-s + (0.758 + 0.651i)11-s + (0.151 + 0.988i)12-s + (−0.954 − 0.299i)13-s + (0.918 − 0.394i)14-s + (0.979 − 0.201i)15-s + (0.820 − 0.571i)16-s + (0.874 − 0.485i)17-s + ⋯ |
L(s) = 1 | + (0.151 + 0.988i)2-s + (0.151 − 0.988i)3-s + (−0.954 + 0.299i)4-s + (0.347 + 0.937i)5-s + 6-s + (−0.250 − 0.968i)7-s + (−0.440 − 0.897i)8-s + (−0.954 − 0.299i)9-s + (−0.874 + 0.485i)10-s + (0.758 + 0.651i)11-s + (0.151 + 0.988i)12-s + (−0.954 − 0.299i)13-s + (0.918 − 0.394i)14-s + (0.979 − 0.201i)15-s + (0.820 − 0.571i)16-s + (0.874 − 0.485i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(311\) |
Sign: | $-0.571 + 0.820i$ |
Analytic conductor: | \(33.4215\) |
Root analytic conductor: | \(33.4215\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{311} (116, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 311,\ (1:\ ),\ -0.571 + 0.820i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.6350252480 + 1.215275606i\) |
\(L(\frac12)\) | \(\approx\) | \(0.6350252480 + 1.215275606i\) |
\(L(1)\) | \(\approx\) | \(0.9446375881 + 0.4037695781i\) |
\(L(1)\) | \(\approx\) | \(0.9446375881 + 0.4037695781i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (0.151 + 0.988i)T \) |
3 | \( 1 + (0.151 - 0.988i)T \) | |
5 | \( 1 + (0.347 + 0.937i)T \) | |
7 | \( 1 + (-0.250 - 0.968i)T \) | |
11 | \( 1 + (0.758 + 0.651i)T \) | |
13 | \( 1 + (-0.954 - 0.299i)T \) | |
17 | \( 1 + (0.874 - 0.485i)T \) | |
19 | \( 1 + (-0.347 + 0.937i)T \) | |
23 | \( 1 + (0.440 + 0.897i)T \) | |
29 | \( 1 + (-0.918 - 0.394i)T \) | |
31 | \( 1 + (0.874 + 0.485i)T \) | |
37 | \( 1 + (0.250 - 0.968i)T \) | |
41 | \( 1 + (0.0506 + 0.998i)T \) | |
43 | \( 1 + (0.250 + 0.968i)T \) | |
47 | \( 1 + (0.918 + 0.394i)T \) | |
53 | \( 1 + (-0.250 - 0.968i)T \) | |
59 | \( 1 + (0.250 + 0.968i)T \) | |
61 | \( 1 + (-0.347 + 0.937i)T \) | |
67 | \( 1 + (-0.0506 + 0.998i)T \) | |
71 | \( 1 + (-0.688 + 0.724i)T \) | |
73 | \( 1 + (-0.612 + 0.790i)T \) | |
79 | \( 1 + (-0.0506 - 0.998i)T \) | |
83 | \( 1 + (0.979 + 0.201i)T \) | |
89 | \( 1 + (-0.250 + 0.968i)T \) | |
97 | \( 1 + (-0.688 + 0.724i)T \) | |
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Imaginary part of the first few zeros on the critical line
−24.678344934063992266606763965653, −23.75468094897443428183622400143, −22.268990145822548031847666086125, −21.976772761901657208318934753452, −21.16841269316600311428500853032, −20.41718452926123105047615752562, −19.46748083021636489047298227968, −18.79690344909833622364404761039, −17.16855325795850169433750136491, −16.83946243162217799218389768901, −15.43747493912137880564329544280, −14.56921160904232951171439455642, −13.63262468499858377364143642554, −12.44544876648774988718487387256, −11.84215483056918596339760786697, −10.704532698534986188545864721139, −9.60128389510756227584320926543, −9.077507019138382911845256835649, −8.36700740477240064554123041883, −6.02302421230632881360570832244, −5.15295279585885045473365153440, −4.32022085889708259262024722059, −3.11768005224015532692933249356, −2.059092634098319288358862949292, −0.41184563506904581442230624808, 1.206605303911243154664003864163, 2.87668209433686295206588135297, 3.99759007071166991772874265927, 5.56611962916022165207167787749, 6.51520110708220259484050898001, 7.351228379794577251531597576389, 7.72537268106445845450786757944, 9.38008955395565536066355286355, 10.13411718096685137218757330283, 11.70023069453312769866012992486, 12.73965860262893637309667913078, 13.61334992921203191742948673468, 14.503058233988130643083818769877, 14.81511721081476623998347064042, 16.46733222448577771459707912736, 17.35400637459172852077126425362, 17.79456146131185050847569709759, 18.9716337505401210881636149170, 19.54583725518563028914029074440, 20.90915176284446549319063602992, 22.25204400386952134730744571209, 23.014818261590459697712653351028, 23.34219125696081418899848019780, 24.70430534431889975557462768238, 25.18951819517932386763284346015