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Properties

Label 1-97-97.90-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.727 - 0.686i$
Analytic cond. $10.4240$
Root an. cond. $10.4240$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.991 − 0.130i)2-s + (−0.793 − 0.608i)3-s + (0.965 − 0.258i)4-s + (0.442 − 0.896i)5-s + (−0.866 − 0.5i)6-s + (−0.997 − 0.0654i)7-s + (0.923 − 0.382i)8-s + (0.258 + 0.965i)9-s + (0.321 − 0.946i)10-s + (0.130 − 0.991i)11-s + (−0.923 − 0.382i)12-s + (−0.896 − 0.442i)13-s + (−0.997 + 0.0654i)14-s + (−0.896 + 0.442i)15-s + (0.866 − 0.5i)16-s + (0.0654 + 0.997i)17-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (−0.793 − 0.608i)3-s + (0.965 − 0.258i)4-s + (0.442 − 0.896i)5-s + (−0.866 − 0.5i)6-s + (−0.997 − 0.0654i)7-s + (0.923 − 0.382i)8-s + (0.258 + 0.965i)9-s + (0.321 − 0.946i)10-s + (0.130 − 0.991i)11-s + (−0.923 − 0.382i)12-s + (−0.896 − 0.442i)13-s + (−0.997 + 0.0654i)14-s + (−0.896 + 0.442i)15-s + (0.866 − 0.5i)16-s + (0.0654 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.727 - 0.686i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7394680427 - 1.861966401i\)
\(L(\frac12)\) \(\approx\) \(0.7394680427 - 1.861966401i\)
\(L(1)\) \(\approx\) \(1.181857926 - 0.8045500074i\)
\(L(1)\) \(\approx\) \(1.181857926 - 0.8045500074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.991 - 0.130i)T \)
3 \( 1 + (-0.793 - 0.608i)T \)
5 \( 1 + (0.442 - 0.896i)T \)
7 \( 1 + (-0.997 - 0.0654i)T \)
11 \( 1 + (0.130 - 0.991i)T \)
13 \( 1 + (-0.896 - 0.442i)T \)
17 \( 1 + (0.0654 + 0.997i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (-0.659 + 0.751i)T \)
29 \( 1 + (-0.321 - 0.946i)T \)
31 \( 1 + (0.608 - 0.793i)T \)
37 \( 1 + (0.751 - 0.659i)T \)
41 \( 1 + (0.946 - 0.321i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.130 + 0.991i)T \)
59 \( 1 + (0.659 + 0.751i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (0.946 + 0.321i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (0.997 - 0.0654i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.07696575928577950876087481362, −29.29763994980695140750211427639, −28.57802758853195883252468124084, −26.98606750777816695950175217433, −25.93546217374554291134436433728, −25.06278091701798458572396318946, −23.52254596897438988086070108938, −22.600546404924384884607155858285, −22.238372969516943920388630773368, −21.17251682111270571794572986185, −19.95303062740384633205700927805, −18.43959491706301876934899672874, −17.08686905766784290286598783944, −16.17357635671320529599963399584, −15.06345709130731348317196631418, −14.20956135858514008871483625233, −12.6554282316719168844483592460, −11.84313865499432549198675457249, −10.44078598839409682305899297579, −9.70652474384704090163158937532, −7.0627752277249415188918920942, −6.42115473961708703993535712671, −5.13747549409034891558387363941, −3.83605379657148212608828888310, −2.439664341836518797120305919507, 0.685053353724443226063156982874, 2.39113046271132271647460648757, 4.241855454909638941442431845635, 5.68627052419977792887160652348, 6.2498844201470327286500007285, 7.80239180708946505302554635943, 9.74445150724100300128223200448, 11.045730596432701619522489757744, 12.30745590487737216360964929192, 12.9901859037328817592137389856, 13.76625638538634238985694547175, 15.53818686476170338480335831511, 16.60421101358818887410701243830, 17.31719173236010325065778741185, 19.21188340997948954879742977221, 19.80599653580181576897585772260, 21.468166155233379205917999946445, 22.0591187683673053639150645060, 23.20442759107795252346753557083, 24.18343413117725666145735423010, 24.76360787968652720238440389392, 25.969550537251106798005264974433, 27.937192895549578522197078127877, 28.7155948943947912090940971906, 29.590735951156529369308444612194

Graph of the $Z$-function along the critical line