Dirichlet series
L(s) = 1 | + (0.986 − 0.164i)2-s + (−0.991 − 0.131i)3-s + (0.945 − 0.324i)4-s + (0.746 − 0.665i)5-s + (−0.999 + 0.0330i)6-s + (−0.652 + 0.757i)7-s + (0.879 − 0.475i)8-s + (0.965 + 0.261i)9-s + (0.627 − 0.778i)10-s + (−0.627 − 0.778i)11-s + (−0.980 + 0.197i)12-s + (−0.846 + 0.533i)13-s + (−0.518 + 0.854i)14-s + (−0.828 + 0.560i)15-s + (0.789 − 0.614i)16-s + (0.340 + 0.940i)17-s + ⋯ |
L(s) = 1 | + (0.986 − 0.164i)2-s + (−0.991 − 0.131i)3-s + (0.945 − 0.324i)4-s + (0.746 − 0.665i)5-s + (−0.999 + 0.0330i)6-s + (−0.652 + 0.757i)7-s + (0.879 − 0.475i)8-s + (0.965 + 0.261i)9-s + (0.627 − 0.778i)10-s + (−0.627 − 0.778i)11-s + (−0.980 + 0.197i)12-s + (−0.846 + 0.533i)13-s + (−0.518 + 0.854i)14-s + (−0.828 + 0.560i)15-s + (0.789 − 0.614i)16-s + (0.340 + 0.940i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(571\) |
Sign: | $0.996 + 0.0874i$ |
Analytic conductor: | \(61.3624\) |
Root analytic conductor: | \(61.3624\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{571} (160, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 571,\ (1:\ ),\ 0.996 + 0.0874i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(3.028935325 + 0.1327418568i\) |
\(L(\frac12)\) | \(\approx\) | \(3.028935325 + 0.1327418568i\) |
\(L(1)\) | \(\approx\) | \(1.602933904 - 0.1652409202i\) |
\(L(1)\) | \(\approx\) | \(1.602933904 - 0.1652409202i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.986 - 0.164i)T \) |
3 | \( 1 + (-0.991 - 0.131i)T \) | |
5 | \( 1 + (0.746 - 0.665i)T \) | |
7 | \( 1 + (-0.652 + 0.757i)T \) | |
11 | \( 1 + (-0.627 - 0.778i)T \) | |
13 | \( 1 + (-0.846 + 0.533i)T \) | |
17 | \( 1 + (0.340 + 0.940i)T \) | |
19 | \( 1 + (0.724 + 0.689i)T \) | |
23 | \( 1 + (0.431 + 0.901i)T \) | |
29 | \( 1 + (0.245 + 0.969i)T \) | |
31 | \( 1 + (0.789 - 0.614i)T \) | |
37 | \( 1 + (0.997 - 0.0660i)T \) | |
41 | \( 1 + (0.401 + 0.915i)T \) | |
43 | \( 1 + (-0.934 - 0.355i)T \) | |
47 | \( 1 + (-0.789 + 0.614i)T \) | |
53 | \( 1 + (0.461 - 0.887i)T \) | |
59 | \( 1 + (-0.0825 + 0.996i)T \) | |
61 | \( 1 + (0.894 + 0.446i)T \) | |
67 | \( 1 + (0.461 - 0.887i)T \) | |
71 | \( 1 + (0.809 + 0.587i)T \) | |
73 | \( 1 + (-0.997 + 0.0660i)T \) | |
79 | \( 1 + (0.574 + 0.818i)T \) | |
83 | \( 1 + (-0.999 + 0.0330i)T \) | |
89 | \( 1 + (0.999 - 0.0330i)T \) | |
97 | \( 1 + (0.601 + 0.799i)T \) | |
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Imaginary part of the first few zeros on the critical line
−22.858964673379294420684389619327, −22.476871770885522668868295411036, −21.64961993722430615931855011750, −20.78797117713087835551797313499, −20.002591743946472361040700644309, −18.748570326215839993113462648660, −17.69406824793679166578774655901, −17.15066133878381309635150666026, −16.20406694413109857994664867026, −15.46045958356413302128647223546, −14.548601475347512853427722279159, −13.53137649981688895494502831477, −12.92826988130228536930367301221, −12.040316304653370324008278837679, −11.081247960997325636103549463784, −10.15512539977175600831930400100, −9.80361489495932559051370050744, −7.53530391086707019690699213708, −7.0015432012823289923647111663, −6.24942549927052768448700863163, −5.17773451195152494328092226656, −4.6230310137426468750338749690, −3.2044400461106634481316303631, −2.35706362722271728046160788959, −0.68994446653608850382994178375, 1.01249458911725458985115662240, 2.07205318971713717336508682987, 3.25165546763163699095041323212, 4.60902402645005010900451930945, 5.49875915063125823919662118031, 5.88404771405257143258207411116, 6.808216176748138659240042475715, 8.08186258002106425013974387261, 9.617167504003941361801634384013, 10.171369460584124968888561070487, 11.376255298299918525174569552260, 12.08777448612803116150051356881, 12.873876202544633749785230729514, 13.33833237869690382455292538444, 14.492128325390657439308342848280, 15.56804204861822940596830411944, 16.43678729808310258665713207168, 16.7829824515314061027747471095, 18.03995871285719740734974779143, 18.96440658789729120597141586300, 19.74070892574076298160581517744, 21.10849199674610866037634139744, 21.54770358584191737168279458494, 22.0637144869986010546797618577, 23.00927815954978216273029057984