Dirichlet series
L(s) = 1 | + (0.995 − 0.0922i)2-s + (0.602 − 0.798i)3-s + (0.982 − 0.183i)4-s + (−0.850 − 0.526i)5-s + (0.526 − 0.850i)6-s + (−0.673 + 0.739i)7-s + (0.961 − 0.273i)8-s + (−0.273 − 0.961i)9-s + (−0.895 − 0.445i)10-s + (0.445 + 0.895i)11-s + (0.445 − 0.895i)12-s + (−0.995 − 0.0922i)13-s + (−0.602 + 0.798i)14-s + (−0.932 + 0.361i)15-s + (0.932 − 0.361i)16-s + (0.850 + 0.526i)17-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0922i)2-s + (0.602 − 0.798i)3-s + (0.982 − 0.183i)4-s + (−0.850 − 0.526i)5-s + (0.526 − 0.850i)6-s + (−0.673 + 0.739i)7-s + (0.961 − 0.273i)8-s + (−0.273 − 0.961i)9-s + (−0.895 − 0.445i)10-s + (0.445 + 0.895i)11-s + (0.445 − 0.895i)12-s + (−0.995 − 0.0922i)13-s + (−0.602 + 0.798i)14-s + (−0.932 + 0.361i)15-s + (0.932 − 0.361i)16-s + (0.850 + 0.526i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(1021\) |
Sign: | $0.870 - 0.493i$ |
Analytic conductor: | \(109.721\) |
Root analytic conductor: | \(109.721\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{1021} (13, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 1021,\ (1:\ ),\ 0.870 - 0.493i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(4.453916324 - 1.174299779i\) |
\(L(\frac12)\) | \(\approx\) | \(4.453916324 - 1.174299779i\) |
\(L(1)\) | \(\approx\) | \(2.065059284 - 0.5379965538i\) |
\(L(1)\) | \(\approx\) | \(2.065059284 - 0.5379965538i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (0.995 - 0.0922i)T \) |
3 | \( 1 + (0.602 - 0.798i)T \) | |
5 | \( 1 + (-0.850 - 0.526i)T \) | |
7 | \( 1 + (-0.673 + 0.739i)T \) | |
11 | \( 1 + (0.445 + 0.895i)T \) | |
13 | \( 1 + (-0.995 - 0.0922i)T \) | |
17 | \( 1 + (0.850 + 0.526i)T \) | |
19 | \( 1 + (0.961 + 0.273i)T \) | |
23 | \( 1 + (0.932 + 0.361i)T \) | |
29 | \( 1 + (-0.739 + 0.673i)T \) | |
31 | \( 1 + (0.961 - 0.273i)T \) | |
37 | \( 1 + (0.798 + 0.602i)T \) | |
41 | \( 1 + (0.273 - 0.961i)T \) | |
43 | \( 1 + (-0.673 + 0.739i)T \) | |
47 | \( 1 + (-0.602 + 0.798i)T \) | |
53 | \( 1 + (0.361 - 0.932i)T \) | |
59 | \( 1 + (0.798 + 0.602i)T \) | |
61 | \( 1 + (0.602 - 0.798i)T \) | |
67 | \( 1 + T \) | |
71 | \( 1 + (-0.273 - 0.961i)T \) | |
73 | \( 1 + (-0.602 - 0.798i)T \) | |
79 | \( 1 + (-0.445 + 0.895i)T \) | |
83 | \( 1 + (0.602 - 0.798i)T \) | |
89 | \( 1 + (-0.982 - 0.183i)T \) | |
97 | \( 1 + (0.673 - 0.739i)T \) | |
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Imaginary part of the first few zeros on the critical line
−21.663579594029502997894948950115, −20.70697292418895610755641958368, −19.99321976454460313070510917761, −19.44210978825407681927394021661, −18.822710894813223904001355923716, −17.02382047821601733876555550329, −16.46215737989774169938628604062, −15.892580207676532970303510704372, −15.0181614220475088839670249676, −14.40060144830185139127292580202, −13.795860221332139977846702344231, −12.94692650631934974336617817439, −11.75147569561682685162677532593, −11.281437274874561765396464606832, −10.28630727023491219581102589771, −9.62175209197452032953733237948, −8.3076575217287368224606643064, −7.42809841123876960969754660886, −6.8862799613290881121940621429, −5.6358459340649853051413999951, −4.66271049523039794969140278173, −3.87383724972541380732716646760, −3.17081837616741189198877126028, −2.65768072902859012568728626340, −0.75101589889473135102802717080, 0.930491236129606882220295693, 1.93656055962469177662378529265, 3.018104252068959110047635306729, 3.583438172591731709977994955615, 4.75237444939306723387861675121, 5.60514382448806107638332714114, 6.68928605774816971368173125141, 7.387121853839215793921604119996, 8.063374408657320474044529533094, 9.27541564601007800845940909600, 9.95472863132661915326664756539, 11.62565878153563173835070978204, 11.9473196059498470470571403271, 12.77288004156731543088110241824, 13.03382462764961647688970628760, 14.387840480179518596079720728637, 14.89254826251470632003037660990, 15.50509304717329873896647514634, 16.472801340345316770268142820505, 17.296540605559853908857998332664, 18.592293765764644122832304963993, 19.36083298950727334488414699042, 19.72973335486567482569382937992, 20.51026044132394459704973439310, 21.22566595708977884958003543311