Dirichlet series
L(s) = 1 | + (−0.695 − 0.718i)2-s + (0.538 − 0.842i)3-s + (−0.0334 + 0.999i)4-s + (0.420 − 0.907i)5-s + (−0.979 + 0.199i)6-s + (−0.645 − 0.763i)7-s + (0.741 − 0.670i)8-s + (−0.420 − 0.907i)9-s + (−0.944 + 0.328i)10-s + (−0.0334 − 0.999i)11-s + (0.824 + 0.565i)12-s + (−0.979 + 0.199i)13-s + (−0.100 + 0.994i)14-s + (−0.538 − 0.842i)15-s + (−0.997 − 0.0667i)16-s + (−0.100 − 0.994i)17-s + ⋯ |
L(s) = 1 | + (−0.695 − 0.718i)2-s + (0.538 − 0.842i)3-s + (−0.0334 + 0.999i)4-s + (0.420 − 0.907i)5-s + (−0.979 + 0.199i)6-s + (−0.645 − 0.763i)7-s + (0.741 − 0.670i)8-s + (−0.420 − 0.907i)9-s + (−0.944 + 0.328i)10-s + (−0.0334 − 0.999i)11-s + (0.824 + 0.565i)12-s + (−0.979 + 0.199i)13-s + (−0.100 + 0.994i)14-s + (−0.538 − 0.842i)15-s + (−0.997 − 0.0667i)16-s + (−0.100 − 0.994i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(283\) |
Sign: | $-0.429 + 0.903i$ |
Analytic conductor: | \(30.4125\) |
Root analytic conductor: | \(30.4125\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{283} (122, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 283,\ (1:\ ),\ -0.429 + 0.903i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(-0.6027707499 - 0.9538798421i\) |
\(L(\frac12)\) | \(\approx\) | \(-0.6027707499 - 0.9538798421i\) |
\(L(1)\) | \(\approx\) | \(0.4400152462 - 0.7188517138i\) |
\(L(1)\) | \(\approx\) | \(0.4400152462 - 0.7188517138i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (-0.695 - 0.718i)T \) |
3 | \( 1 + (0.538 - 0.842i)T \) | |
5 | \( 1 + (0.420 - 0.907i)T \) | |
7 | \( 1 + (-0.645 - 0.763i)T \) | |
11 | \( 1 + (-0.0334 - 0.999i)T \) | |
13 | \( 1 + (-0.979 + 0.199i)T \) | |
17 | \( 1 + (-0.100 - 0.994i)T \) | |
19 | \( 1 + (0.979 - 0.199i)T \) | |
23 | \( 1 + (0.964 + 0.264i)T \) | |
29 | \( 1 + (0.359 + 0.933i)T \) | |
31 | \( 1 + (0.892 + 0.451i)T \) | |
37 | \( 1 + (-0.860 - 0.509i)T \) | |
41 | \( 1 + (-0.741 - 0.670i)T \) | |
43 | \( 1 + (-0.480 - 0.876i)T \) | |
47 | \( 1 + (0.979 + 0.199i)T \) | |
53 | \( 1 + (0.997 - 0.0667i)T \) | |
59 | \( 1 + (-0.166 + 0.986i)T \) | |
61 | \( 1 + (-0.944 - 0.328i)T \) | |
67 | \( 1 + (0.824 - 0.565i)T \) | |
71 | \( 1 + (-0.0334 - 0.999i)T \) | |
73 | \( 1 + (-0.892 + 0.451i)T \) | |
79 | \( 1 + (-0.480 + 0.876i)T \) | |
83 | \( 1 + (0.991 + 0.133i)T \) | |
89 | \( 1 + (0.480 + 0.876i)T \) | |
97 | \( 1 + (0.964 + 0.264i)T \) | |
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Imaginary part of the first few zeros on the critical line
−26.26651083335681487559953066989, −25.10664621716583728287362396099, −24.81978096744031310453189404379, −23.03292780564319806082024681914, −22.44786550384200313260497655318, −21.587417979993199350787096615832, −20.340593619569320076182814799719, −19.378806712350284348355968818221, −18.754049882941785755112478316277, −17.58624186418008115562484216886, −16.85603377642277029949208756194, −15.50285580315283378492935433168, −15.17378839913311825409298281363, −14.409797244718033927754607067, −13.269005715807695337077322042820, −11.6788864409050497364319848654, −10.17453124078595970825867714137, −10.00135057363322891761009546030, −9.02651093784371890578695661866, −7.86192594899353163896640764127, −6.831385616681818143183132059173, −5.7500103094180832525702585779, −4.69561762232175967646359803469, −3.01501511907073385989258060370, −2.06277557937155880756626070319, 0.436342422898845606144600318275, 1.174846587807873413093595160849, 2.606109167418416195216839128318, 3.5190223146992203834598157000, 5.10453838912333480888755388285, 6.81142358836761366480718426451, 7.537820073963375047603943570327, 8.79634978084904900852919856683, 9.30375225666765775153966272987, 10.39898219482820730989467597618, 11.81077163556741257583210284886, 12.46807587945913207461961728450, 13.64533902060202154723576926790, 13.782861496959308139029100892463, 15.86290192718025025893326364012, 16.79742798096948683953108494916, 17.46295397015266412207828368827, 18.52763259196656122172771681896, 19.42473077763621591153839473841, 20.02631829525606128174414621622, 20.753702819559980450677527710949, 21.750925914529500937987859514514, 22.904653001350589550332376215324, 24.158497276375796957438227748357, 24.833355981195081044502064475015