Dirichlet series
L(s) = 1 | + (−0.758 − 0.651i)2-s + (−0.758 + 0.651i)3-s + (0.151 + 0.988i)4-s + (0.820 + 0.571i)5-s + 6-s + (−0.612 + 0.790i)7-s + (0.528 − 0.848i)8-s + (0.151 − 0.988i)9-s + (−0.250 − 0.968i)10-s + (−0.347 + 0.937i)11-s + (−0.758 − 0.651i)12-s + (0.151 − 0.988i)13-s + (0.979 − 0.201i)14-s + (−0.994 + 0.101i)15-s + (−0.954 + 0.299i)16-s + (0.250 + 0.968i)17-s + ⋯ |
L(s) = 1 | + (−0.758 − 0.651i)2-s + (−0.758 + 0.651i)3-s + (0.151 + 0.988i)4-s + (0.820 + 0.571i)5-s + 6-s + (−0.612 + 0.790i)7-s + (0.528 − 0.848i)8-s + (0.151 − 0.988i)9-s + (−0.250 − 0.968i)10-s + (−0.347 + 0.937i)11-s + (−0.758 − 0.651i)12-s + (0.151 − 0.988i)13-s + (0.979 − 0.201i)14-s + (−0.994 + 0.101i)15-s + (−0.954 + 0.299i)16-s + (0.250 + 0.968i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(311\) |
Sign: | $-0.887 - 0.460i$ |
Analytic conductor: | \(33.4215\) |
Root analytic conductor: | \(33.4215\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{311} (220, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 311,\ (1:\ ),\ -0.887 - 0.460i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(-0.06616803983 + 0.2712253485i\) |
\(L(\frac12)\) | \(\approx\) | \(-0.06616803983 + 0.2712253485i\) |
\(L(1)\) | \(\approx\) | \(0.5103146662 + 0.1542597265i\) |
\(L(1)\) | \(\approx\) | \(0.5103146662 + 0.1542597265i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (-0.758 - 0.651i)T \) |
3 | \( 1 + (-0.758 + 0.651i)T \) | |
5 | \( 1 + (0.820 + 0.571i)T \) | |
7 | \( 1 + (-0.612 + 0.790i)T \) | |
11 | \( 1 + (-0.347 + 0.937i)T \) | |
13 | \( 1 + (0.151 - 0.988i)T \) | |
17 | \( 1 + (0.250 + 0.968i)T \) | |
19 | \( 1 + (-0.820 + 0.571i)T \) | |
23 | \( 1 + (-0.528 + 0.848i)T \) | |
29 | \( 1 + (-0.979 - 0.201i)T \) | |
31 | \( 1 + (0.250 - 0.968i)T \) | |
37 | \( 1 + (0.612 + 0.790i)T \) | |
41 | \( 1 + (-0.688 + 0.724i)T \) | |
43 | \( 1 + (0.612 - 0.790i)T \) | |
47 | \( 1 + (0.979 + 0.201i)T \) | |
53 | \( 1 + (-0.612 + 0.790i)T \) | |
59 | \( 1 + (0.612 - 0.790i)T \) | |
61 | \( 1 + (-0.820 + 0.571i)T \) | |
67 | \( 1 + (0.688 + 0.724i)T \) | |
71 | \( 1 + (-0.918 + 0.394i)T \) | |
73 | \( 1 + (-0.440 - 0.897i)T \) | |
79 | \( 1 + (0.688 - 0.724i)T \) | |
83 | \( 1 + (-0.994 - 0.101i)T \) | |
89 | \( 1 + (-0.612 - 0.790i)T \) | |
97 | \( 1 + (-0.918 + 0.394i)T \) | |
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Imaginary part of the first few zeros on the critical line
−24.397928963376631059165232512143, −23.894654062340149611560639627290, −23.074936092586366614985449055053, −21.98097417356215825370763014986, −20.816418043537974497779538410813, −19.687067363603625347465358166499, −18.839778818181072093202594954936, −18.08911431681421859047055068954, −17.1138637095415059349898897690, −16.48944047688587874492912552460, −16.04988534739322394061255100592, −14.12901297533178452826497803598, −13.65374596176268540296428726316, −12.62419327700578431069649905467, −11.219905930406515368894994763786, −10.46260215764079425311549605128, −9.41027390247972717225290447627, −8.452130494306595763547343630700, −7.19654700037837547021357595986, −6.45590927692030964555463265947, −5.64608641498914906510328155423, −4.55990623145062181215695519693, −2.29887035110136441795346496033, −1.0293283846891661048706384008, −0.135510942139527303091972956651, 1.71451534227383525120891463327, 2.87815572986456323067157019702, 4.00430738259928686312646491325, 5.61078583857283897902765648384, 6.33454591928848083857715013776, 7.70857205212652500110376522360, 9.07102312264631443545942839876, 10.02275961077327894548776389491, 10.30100351861444245770219839392, 11.44603688805371182413608337753, 12.515955857145587510269112140269, 13.11630228407000628737828113026, 15.00097027190658107988591219735, 15.535049547987703915384462895530, 16.863997521108950391250688938985, 17.44028768262753156174366160668, 18.285299895198248381138447088140, 18.97670117445343620476837573583, 20.34776652509334223154707825761, 21.069782569634151381291159657, 22.03759975224951231576936271018, 22.34685706330433859124735581041, 23.45116478249365985944410823001, 25.187998659060017083245914055425, 25.69170645048087334849334190404