Dirichlet series
L(s) = 1 | + (0.614 + 0.789i)2-s + (0.945 + 0.324i)3-s + (−0.245 + 0.969i)4-s + (−0.546 + 0.837i)5-s + (0.324 + 0.945i)6-s + (0.735 + 0.677i)7-s + (−0.915 + 0.401i)8-s + (0.789 + 0.614i)9-s + (−0.996 + 0.0825i)10-s + (0.986 − 0.164i)11-s + (−0.546 + 0.837i)12-s + (0.837 + 0.546i)13-s + (−0.0825 + 0.996i)14-s + (−0.789 + 0.614i)15-s + (−0.879 − 0.475i)16-s + (0.546 − 0.837i)17-s + ⋯ |
L(s) = 1 | + (0.614 + 0.789i)2-s + (0.945 + 0.324i)3-s + (−0.245 + 0.969i)4-s + (−0.546 + 0.837i)5-s + (0.324 + 0.945i)6-s + (0.735 + 0.677i)7-s + (−0.915 + 0.401i)8-s + (0.789 + 0.614i)9-s + (−0.996 + 0.0825i)10-s + (0.986 − 0.164i)11-s + (−0.546 + 0.837i)12-s + (0.837 + 0.546i)13-s + (−0.0825 + 0.996i)14-s + (−0.789 + 0.614i)15-s + (−0.879 − 0.475i)16-s + (0.546 − 0.837i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(229\) |
Sign: | $-0.930 + 0.365i$ |
Analytic conductor: | \(24.6094\) |
Root analytic conductor: | \(24.6094\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{229} (177, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 229,\ (1:\ ),\ -0.930 + 0.365i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.6983256800 + 3.686414959i\) |
\(L(\frac12)\) | \(\approx\) | \(0.6983256800 + 3.686414959i\) |
\(L(1)\) | \(\approx\) | \(1.325260209 + 1.571497185i\) |
\(L(1)\) | \(\approx\) | \(1.325260209 + 1.571497185i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 229 | \( 1 \) |
good | 2 | \( 1 + (0.614 + 0.789i)T \) |
3 | \( 1 + (0.945 + 0.324i)T \) | |
5 | \( 1 + (-0.546 + 0.837i)T \) | |
7 | \( 1 + (0.735 + 0.677i)T \) | |
11 | \( 1 + (0.986 - 0.164i)T \) | |
13 | \( 1 + (0.837 + 0.546i)T \) | |
17 | \( 1 + (0.546 - 0.837i)T \) | |
19 | \( 1 + (0.546 + 0.837i)T \) | |
23 | \( 1 + (-0.996 - 0.0825i)T \) | |
29 | \( 1 + (-0.735 - 0.677i)T \) | |
31 | \( 1 + (-0.164 - 0.986i)T \) | |
37 | \( 1 + (-0.879 + 0.475i)T \) | |
41 | \( 1 + (-0.614 - 0.789i)T \) | |
43 | \( 1 + (-0.879 + 0.475i)T \) | |
47 | \( 1 + (0.614 - 0.789i)T \) | |
53 | \( 1 + (0.945 + 0.324i)T \) | |
59 | \( 1 + (0.475 - 0.879i)T \) | |
61 | \( 1 + (0.789 + 0.614i)T \) | |
67 | \( 1 + (0.614 - 0.789i)T \) | |
71 | \( 1 + (0.986 + 0.164i)T \) | |
73 | \( 1 + (0.915 - 0.401i)T \) | |
79 | \( 1 + (-0.735 + 0.677i)T \) | |
83 | \( 1 + (-0.879 - 0.475i)T \) | |
89 | \( 1 + iT \) | |
97 | \( 1 + (0.401 - 0.915i)T \) | |
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Imaginary part of the first few zeros on the critical line
−25.580718068148825714268514895852, −24.449337898043887378401465316663, −23.938268863328340558978402165736, −23.11950085294710665567408288567, −21.74018910459864240032561832116, −20.82383894072114569871044430491, −19.98979708786888966547175342388, −19.800436142372053815302097112094, −18.50328411469630664389243922473, −17.44437541234093530156484221026, −15.9443727539384874643310373812, −14.91683451953057538235914235584, −14.09366721098991259668266363346, −13.24655033523551422439499365459, −12.35535553315515017728689597000, −11.4358601630122642480134010271, −10.21793587994124069708333869852, −9.00389792750985601460451709108, −8.21517263221918107250965181580, −6.925941587276886542246230671596, −5.312799490268040406611056458639, −4.03614201600572887250910127065, −3.52445140315355209299337163643, −1.68011810780745770238686785102, −0.99314182253582732530130007195, 2.105249795409076575681458308906, 3.48674351451583502838529192656, 4.08794074621974456722302409942, 5.5573059655227872923531071779, 6.816476342287124017228595323, 7.85609865178377822618731325178, 8.57447558724299762251027504411, 9.71745326315519545419976002471, 11.4161741387893112995611597568, 12.05727669379593738833537256815, 13.77373723382384459983227932330, 14.216567666548098905589051746932, 15.0509157506100224960813578197, 15.78022945989149627536111350979, 16.75582363381928634452397930044, 18.35378088502660181104468818272, 18.76040343594886966798922353992, 20.27506688125594125041689265086, 21.0946997835425549170580872096, 22.049536250389949542685630565710, 22.71608285634270282432090873721, 23.97644625509057845877099142378, 24.74690692865486533202916678014, 25.5579534999849699521969315407, 26.36898294221623403889828261470