Dirichlet series
L(s) = 1 | + (−0.969 − 0.245i)2-s + (0.789 + 0.614i)3-s + (0.879 + 0.475i)4-s + (0.401 + 0.915i)5-s + (−0.614 − 0.789i)6-s + (0.996 − 0.0825i)7-s + (−0.735 − 0.677i)8-s + (0.245 + 0.969i)9-s + (−0.164 − 0.986i)10-s + (−0.945 + 0.324i)11-s + (0.401 + 0.915i)12-s + (−0.915 + 0.401i)13-s + (−0.986 − 0.164i)14-s + (−0.245 + 0.969i)15-s + (0.546 + 0.837i)16-s + (−0.401 − 0.915i)17-s + ⋯ |
L(s) = 1 | + (−0.969 − 0.245i)2-s + (0.789 + 0.614i)3-s + (0.879 + 0.475i)4-s + (0.401 + 0.915i)5-s + (−0.614 − 0.789i)6-s + (0.996 − 0.0825i)7-s + (−0.735 − 0.677i)8-s + (0.245 + 0.969i)9-s + (−0.164 − 0.986i)10-s + (−0.945 + 0.324i)11-s + (0.401 + 0.915i)12-s + (−0.915 + 0.401i)13-s + (−0.986 − 0.164i)14-s + (−0.245 + 0.969i)15-s + (0.546 + 0.837i)16-s + (−0.401 − 0.915i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(229\) |
Sign: | $-0.847 + 0.531i$ |
Analytic conductor: | \(24.6094\) |
Root analytic conductor: | \(24.6094\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{229} (128, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 229,\ (1:\ ),\ -0.847 + 0.531i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.3448470510 + 1.199251110i\) |
\(L(\frac12)\) | \(\approx\) | \(0.3448470510 + 1.199251110i\) |
\(L(1)\) | \(\approx\) | \(0.8156518261 + 0.4155666596i\) |
\(L(1)\) | \(\approx\) | \(0.8156518261 + 0.4155666596i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 229 | \( 1 \) |
good | 2 | \( 1 + (-0.969 - 0.245i)T \) |
3 | \( 1 + (0.789 + 0.614i)T \) | |
5 | \( 1 + (0.401 + 0.915i)T \) | |
7 | \( 1 + (0.996 - 0.0825i)T \) | |
11 | \( 1 + (-0.945 + 0.324i)T \) | |
13 | \( 1 + (-0.915 + 0.401i)T \) | |
17 | \( 1 + (-0.401 - 0.915i)T \) | |
19 | \( 1 + (-0.401 + 0.915i)T \) | |
23 | \( 1 + (-0.164 + 0.986i)T \) | |
29 | \( 1 + (-0.996 + 0.0825i)T \) | |
31 | \( 1 + (-0.324 - 0.945i)T \) | |
37 | \( 1 + (0.546 - 0.837i)T \) | |
41 | \( 1 + (0.969 + 0.245i)T \) | |
43 | \( 1 + (0.546 - 0.837i)T \) | |
47 | \( 1 + (-0.969 + 0.245i)T \) | |
53 | \( 1 + (0.789 + 0.614i)T \) | |
59 | \( 1 + (0.837 - 0.546i)T \) | |
61 | \( 1 + (0.245 + 0.969i)T \) | |
67 | \( 1 + (-0.969 + 0.245i)T \) | |
71 | \( 1 + (-0.945 - 0.324i)T \) | |
73 | \( 1 + (0.735 + 0.677i)T \) | |
79 | \( 1 + (-0.996 - 0.0825i)T \) | |
83 | \( 1 + (0.546 + 0.837i)T \) | |
89 | \( 1 - iT \) | |
97 | \( 1 + (0.677 + 0.735i)T \) | |
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Imaginary part of the first few zeros on the critical line
−25.836460245165852365080291781941, −24.70952086898258880233647040084, −24.27932936980733252864615492420, −23.706298586510878423614394046354, −21.543525923484555201946543054189, −20.79513341595733323906272248931, −20.02959843026343052680811046688, −19.2199344564375907332083019779, −18.00546327496322175271788205533, −17.59975871242822275139304473554, −16.47615804354175014992072252135, −15.22479695781311381182399486098, −14.544488210893518373919960759499, −13.21356030048478951447113499956, −12.370560125654734812417073552457, −11.03877957190088901146590293240, −9.88808354479531875753039534036, −8.7218780328907387463973175477, −8.254208709182664217831574172606, −7.31621730092392414883384610064, −5.941796616513511470219113342, −4.73885215354260548519602495914, −2.59558440678386727866286531507, −1.75455570014505435981161077828, −0.45902883128581555000782019746, 1.98763650272810497431078053490, 2.55401286827627439538135822835, 3.99985439644172129862052578359, 5.52394653092429328395238046033, 7.40047362853785185263439188161, 7.69358778219467319972235288848, 9.13225346844499813375455806906, 9.91854265107097839306811665982, 10.75614696889498107618704192978, 11.59527127165444032769443930313, 13.20789675287864812448043426299, 14.469093983026727394991249507359, 15.05060340390296836024693862617, 16.13071366593393435503162553425, 17.28215222191140940978522482045, 18.19622572343720096695400673113, 18.92920104830344020661865373000, 19.99464661812533848649390741560, 20.94013076070689712383721126559, 21.4117291516035781248159447725, 22.49121453952134186541820451792, 24.09305215575971567441782086942, 25.05552850304801213984906525665, 25.85384535140517261698378925713, 26.634897779250262243060639990133