Dirichlet series
L(s) = 1 | + (−0.387 − 0.921i)2-s + (0.438 + 0.898i)3-s + (−0.699 + 0.714i)4-s + (−0.666 − 0.745i)5-s + (0.658 − 0.752i)6-s + (−0.988 + 0.154i)7-s + (0.930 + 0.367i)8-s + (−0.616 + 0.787i)9-s + (−0.428 + 0.903i)10-s + (0.955 + 0.294i)11-s + (−0.948 − 0.315i)12-s + (0.0442 − 0.999i)13-s + (0.525 + 0.850i)14-s + (0.377 − 0.925i)15-s + (−0.0221 − 0.999i)16-s + (−0.506 + 0.862i)17-s + ⋯ |
L(s) = 1 | + (−0.387 − 0.921i)2-s + (0.438 + 0.898i)3-s + (−0.699 + 0.714i)4-s + (−0.666 − 0.745i)5-s + (0.658 − 0.752i)6-s + (−0.988 + 0.154i)7-s + (0.930 + 0.367i)8-s + (−0.616 + 0.787i)9-s + (−0.428 + 0.903i)10-s + (0.955 + 0.294i)11-s + (−0.948 − 0.315i)12-s + (0.0442 − 0.999i)13-s + (0.525 + 0.850i)14-s + (0.377 − 0.925i)15-s + (−0.0221 − 0.999i)16-s + (−0.506 + 0.862i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(569\) |
Sign: | $0.710 - 0.703i$ |
Analytic conductor: | \(61.1475\) |
Root analytic conductor: | \(61.1475\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{569} (157, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 569,\ (1:\ ),\ 0.710 - 0.703i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.8223572548 - 0.3383514429i\) |
\(L(\frac12)\) | \(\approx\) | \(0.8223572548 - 0.3383514429i\) |
\(L(1)\) | \(\approx\) | \(0.6887492664 - 0.1306201796i\) |
\(L(1)\) | \(\approx\) | \(0.6887492664 - 0.1306201796i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 569 | \( 1 \) |
good | 2 | \( 1 + (-0.387 - 0.921i)T \) |
3 | \( 1 + (0.438 + 0.898i)T \) | |
5 | \( 1 + (-0.666 - 0.745i)T \) | |
7 | \( 1 + (-0.988 + 0.154i)T \) | |
11 | \( 1 + (0.955 + 0.294i)T \) | |
13 | \( 1 + (0.0442 - 0.999i)T \) | |
17 | \( 1 + (-0.506 + 0.862i)T \) | |
19 | \( 1 + (-0.999 - 0.0110i)T \) | |
23 | \( 1 + (-0.820 - 0.571i)T \) | |
29 | \( 1 + (0.967 + 0.251i)T \) | |
31 | \( 1 + (-0.999 - 0.0331i)T \) | |
37 | \( 1 + (0.589 - 0.807i)T \) | |
41 | \( 1 + (-0.984 + 0.176i)T \) | |
43 | \( 1 + (0.921 + 0.387i)T \) | |
47 | \( 1 + (-0.867 - 0.496i)T \) | |
53 | \( 1 + (0.752 + 0.658i)T \) | |
59 | \( 1 + (-0.496 + 0.867i)T \) | |
61 | \( 1 + (0.580 + 0.814i)T \) | |
67 | \( 1 + (-0.506 - 0.862i)T \) | |
71 | \( 1 + (-0.930 + 0.367i)T \) | |
73 | \( 1 + (0.0110 - 0.999i)T \) | |
79 | \( 1 + (0.132 - 0.991i)T \) | |
83 | \( 1 + (0.888 + 0.457i)T \) | |
89 | \( 1 + (-0.0331 - 0.999i)T \) | |
97 | \( 1 + (0.780 - 0.624i)T \) | |
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Imaginary part of the first few zeros on the critical line
−23.50855235242290748340686902177, −22.576007625799306976033866889759, −21.906946049146468907479591071455, −20.11338216043621025924524222278, −19.44565935978512020649731002641, −19.01167791335219581566995925583, −18.27198851425485516333126063669, −17.314034389452670180202779528410, −16.38629181993467821783553653508, −15.61662890589388924201404814507, −14.61059771833614087272061964776, −14.02416926267951013379025015068, −13.262805343888066945977008911758, −12.0782491266298538152035191521, −11.21343438077114636119790344033, −9.879797252413444272642021271451, −9.05881640492692711419597753656, −8.2473854830451123732048808153, −7.15094740152775475826367452273, −6.67950463745208266786836143035, −6.10122398717638037514182186035, −4.289331098737028320629743184588, −3.42432121926597888498204521789, −2.03817161527280965979263909039, −0.57385185021322918292074080112, 0.4351671326208460558655815583, 1.99650773945515032305948149974, 3.21274787268320531631561099108, 3.96686150362264210643288215416, 4.61292624957655477951588706882, 6.01902498011407796072885306051, 7.60847886221758698002048198761, 8.687231411073003075731787475258, 8.95870641047407039395155577845, 10.08387312475106533216125466812, 10.66876162030071212050707360572, 11.81882674750274706289181600407, 12.64246861998459192417204865126, 13.25338497417020526691070878731, 14.57424832494533196387863058431, 15.429943566609427338424412075394, 16.40951099230861480887632879173, 16.902686345561403154745548850273, 17.98012475677503769910989945257, 19.368879316191943615023628618865, 19.72635928833609981177949715617, 20.176748096082794141384314651, 21.18514521016286505987177801558, 22.020449933040401164295817615496, 22.59751180820291584889959533535