Dirichlet series
L(s) = 1 | + (0.979 − 0.199i)2-s + (−0.0111 − 0.999i)3-s + (0.920 − 0.390i)4-s + (−0.519 + 0.854i)5-s + (−0.210 − 0.977i)6-s + (0.999 + 0.0445i)7-s + (0.824 − 0.565i)8-s + (−0.999 + 0.0222i)9-s + (−0.338 + 0.940i)10-s + (−0.122 − 0.992i)11-s + (−0.400 − 0.916i)12-s + (0.951 + 0.306i)13-s + (0.987 − 0.155i)14-s + (0.860 + 0.509i)15-s + (0.695 − 0.718i)16-s + (−0.628 + 0.777i)17-s + ⋯ |
L(s) = 1 | + (0.979 − 0.199i)2-s + (−0.0111 − 0.999i)3-s + (0.920 − 0.390i)4-s + (−0.519 + 0.854i)5-s + (−0.210 − 0.977i)6-s + (0.999 + 0.0445i)7-s + (0.824 − 0.565i)8-s + (−0.999 + 0.0222i)9-s + (−0.338 + 0.940i)10-s + (−0.122 − 0.992i)11-s + (−0.400 − 0.916i)12-s + (0.951 + 0.306i)13-s + (0.987 − 0.155i)14-s + (0.860 + 0.509i)15-s + (0.695 − 0.718i)16-s + (−0.628 + 0.777i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(283\) |
Sign: | $0.0408 - 0.999i$ |
Analytic conductor: | \(30.4125\) |
Root analytic conductor: | \(30.4125\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{283} (124, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 283,\ (1:\ ),\ 0.0408 - 0.999i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(2.769625329 - 2.658725822i\) |
\(L(\frac12)\) | \(\approx\) | \(2.769625329 - 2.658725822i\) |
\(L(1)\) | \(\approx\) | \(1.864152992 - 0.8845261308i\) |
\(L(1)\) | \(\approx\) | \(1.864152992 - 0.8845261308i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (0.979 - 0.199i)T \) |
3 | \( 1 + (-0.0111 - 0.999i)T \) | |
5 | \( 1 + (-0.519 + 0.854i)T \) | |
7 | \( 1 + (0.999 + 0.0445i)T \) | |
11 | \( 1 + (-0.122 - 0.992i)T \) | |
13 | \( 1 + (0.951 + 0.306i)T \) | |
17 | \( 1 + (-0.628 + 0.777i)T \) | |
19 | \( 1 + (0.741 - 0.670i)T \) | |
23 | \( 1 + (0.441 - 0.897i)T \) | |
29 | \( 1 + (-0.296 - 0.955i)T \) | |
31 | \( 1 + (0.929 - 0.369i)T \) | |
37 | \( 1 + (0.610 + 0.791i)T \) | |
41 | \( 1 + (0.902 - 0.431i)T \) | |
43 | \( 1 + (-0.964 + 0.264i)T \) | |
47 | \( 1 + (0.210 - 0.977i)T \) | |
53 | \( 1 + (-0.695 - 0.718i)T \) | |
59 | \( 1 + (-0.575 + 0.818i)T \) | |
61 | \( 1 + (-0.645 + 0.763i)T \) | |
67 | \( 1 + (-0.593 - 0.805i)T \) | |
71 | \( 1 + (0.920 + 0.390i)T \) | |
73 | \( 1 + (-0.929 - 0.369i)T \) | |
79 | \( 1 + (-0.964 - 0.264i)T \) | |
83 | \( 1 + (0.882 + 0.470i)T \) | |
89 | \( 1 + (-0.711 - 0.703i)T \) | |
97 | \( 1 + (0.556 + 0.830i)T \) | |
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Imaginary part of the first few zeros on the critical line
−25.26573618554630050153849025575, −24.7116668594379403259186394759, −23.39071230612727700764532958879, −23.104542835513630530894435047625, −21.92937395283812238932912202158, −20.900157315774985572242226015573, −20.53446040439604353340983053602, −19.865780281089841573136770528291, −17.93911409178016003450966191127, −17.02999956064573847257492474858, −15.95518761062819649470361246664, −15.54898116566920032952043224422, −14.56988467813496486227682696261, −13.64348452058675070999584300848, −12.46447693348036356980512489899, −11.51523986334943308792599499198, −10.88227003047883059479222189665, −9.42731308545054988184964997762, −8.28098942896346046620040719746, −7.407667146790902866620063694501, −5.71258045193436201064068304999, −4.86959505440246565516550303790, −4.28492443366299107467372796666, −3.14299385242448693541417009562, −1.47263312117357575251585482597, 0.913391974525250675562297981634, 2.23323559171437152144088115136, 3.22225637069164582270291638180, 4.46555246409204753270138052003, 5.90690579022073715957048847994, 6.58087234925165447111536903404, 7.69337947803337471760462121384, 8.530721049574891246936545699875, 10.674813285029522063588368346977, 11.33743613752504794083620544540, 11.83441850180718226268142106757, 13.275654473418157010083966355928, 13.80646087938417008161488744080, 14.736483121792107849608050611463, 15.547770421366471096136234693980, 16.833646413775696560426143617870, 18.12267789716323572225376448176, 18.83804278690482489135197915326, 19.64442679378264191922314472915, 20.6713334088671008554592487764, 21.63969306853881722645190063890, 22.59428200549920095294304262315, 23.39441525072983012101373533248, 24.17863812157396059436290758253, 24.60784795326696716429829294099