Dirichlet series
L(s) = 1 | + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + i·10-s + (−0.309 + 0.951i)11-s + (0.587 − 0.809i)12-s + i·13-s + (0.951 + 0.309i)14-s + (0.587 + 0.809i)15-s + (−0.809 − 0.587i)16-s − i·17-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + i·10-s + (−0.309 + 0.951i)11-s + (0.587 − 0.809i)12-s + i·13-s + (0.951 + 0.309i)14-s + (0.587 + 0.809i)15-s + (−0.809 − 0.587i)16-s − i·17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(1061\) |
Sign: | $0.967 + 0.252i$ |
Analytic conductor: | \(114.020\) |
Root analytic conductor: | \(114.020\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{1061} (29, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 1061,\ (1:\ ),\ 0.967 + 0.252i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(0.3711650602 + 0.04754367122i\) |
\(L(\frac12)\) | \(\approx\) | \(0.3711650602 + 0.04754367122i\) |
\(L(1)\) | \(\approx\) | \(0.3994632673 - 0.1353694803i\) |
\(L(1)\) | \(\approx\) | \(0.3994632673 - 0.1353694803i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 1061 | \( 1 \) |
good | 2 | \( 1 + (-0.587 - 0.809i)T \) |
3 | \( 1 + (-0.951 - 0.309i)T \) | |
5 | \( 1 + (-0.809 - 0.587i)T \) | |
7 | \( 1 + (-0.809 + 0.587i)T \) | |
11 | \( 1 + (-0.309 + 0.951i)T \) | |
13 | \( 1 + iT \) | |
17 | \( 1 - iT \) | |
19 | \( 1 + (0.309 + 0.951i)T \) | |
23 | \( 1 + (-0.309 - 0.951i)T \) | |
29 | \( 1 + (0.587 - 0.809i)T \) | |
31 | \( 1 - T \) | |
37 | \( 1 + T \) | |
41 | \( 1 + (-0.309 - 0.951i)T \) | |
43 | \( 1 + (0.587 - 0.809i)T \) | |
47 | \( 1 + (0.309 + 0.951i)T \) | |
53 | \( 1 + (-0.809 - 0.587i)T \) | |
59 | \( 1 + iT \) | |
61 | \( 1 + iT \) | |
67 | \( 1 + (-0.309 - 0.951i)T \) | |
71 | \( 1 + (0.587 + 0.809i)T \) | |
73 | \( 1 + iT \) | |
79 | \( 1 + (-0.587 - 0.809i)T \) | |
83 | \( 1 - T \) | |
89 | \( 1 + (-0.951 - 0.309i)T \) | |
97 | \( 1 + (0.809 - 0.587i)T \) | |
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Imaginary part of the first few zeros on the critical line
−21.71664729836442597320510997051, −20.035696266460054091747297886, −19.682529805704388080332525928886, −18.72003144533357122354772734134, −18.07630036046461872845600959585, −17.30178176780979197116456853506, −16.50089367907609529478356467639, −15.8683094558784402773076466651, −15.42003345585032527979534342890, −14.515114138142286605182033934542, −13.37566452090731162832008139230, −12.66064322973349549223036950948, −11.28860867401931111622049905283, −10.84239364828179675006573970697, −10.18102105686094135452473146743, −9.3084425953096735624040569227, −8.119937460084355776590204825897, −7.4318489894710816528711175368, −6.5686203862746824067097043655, −5.95917193296480871493801268327, −5.037776178276920625828772046009, −3.93392103713000093367517101323, −3.10607283646905600184430153802, −1.07591073475388138663382013282, −0.25568356732185042453508063916, 0.515978302151764303375464414095, 1.69947115942990444425587456843, 2.65558037504655114622033158730, 4.03510037020800512352890983788, 4.59772191282846897395292500305, 5.719678147062589764809618928570, 6.96831262595688933363125126898, 7.510779109403383798931323545480, 8.57586833323029315897762090608, 9.4921142797832644751384682031, 10.10001932885228647305514696298, 11.19386778127461807610202506303, 11.905629096665871092705822081545, 12.384573383885446942602111791032, 12.873625773150515896125030252341, 13.987256917920423148215433327567, 15.490604293240323942171904949934, 16.26796676720351789226236380914, 16.5623511649533718645394882902, 17.570670329100422352591011263699, 18.557514136550808425557793211, 18.767419771697007242126430859095, 19.68975133561218876984911875825, 20.51687652330426779162816526158, 21.18591413282756639360040454225