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Properties

Label 1-2557-2557.1034-r0-0-0
Degree $1$
Conductor $2557$
Sign $0.151 + 0.988i$
Analytic cond. $11.8746$
Root an. cond. $11.8746$
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  + (0.695 − 0.718i)2-s + (−0.946 + 0.323i)3-s + (−0.0319 − 0.999i)4-s + (−0.825 − 0.564i)5-s + (−0.425 + 0.904i)6-s + (−0.936 − 0.351i)7-s + (−0.740 − 0.672i)8-s + (0.790 − 0.612i)9-s + (−0.979 + 0.200i)10-s + (0.0466 + 0.998i)11-s + (0.353 + 0.935i)12-s + (0.974 + 0.224i)13-s + (−0.903 + 0.428i)14-s + (0.963 + 0.267i)15-s + (−0.997 + 0.0638i)16-s + (0.958 + 0.286i)17-s + ⋯
L(s)  = 1  + (0.695 − 0.718i)2-s + (−0.946 + 0.323i)3-s + (−0.0319 − 0.999i)4-s + (−0.825 − 0.564i)5-s + (−0.425 + 0.904i)6-s + (−0.936 − 0.351i)7-s + (−0.740 − 0.672i)8-s + (0.790 − 0.612i)9-s + (−0.979 + 0.200i)10-s + (0.0466 + 0.998i)11-s + (0.353 + 0.935i)12-s + (0.974 + 0.224i)13-s + (−0.903 + 0.428i)14-s + (0.963 + 0.267i)15-s + (−0.997 + 0.0638i)16-s + (0.958 + 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(11.8746\)
Root analytic conductor: \(11.8746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (0:\ ),\ 0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07820509164 - 0.06716394735i\)
\(L(\frac12)\) \(\approx\) \(-0.07820509164 - 0.06716394735i\)
\(L(1)\) \(\approx\) \(0.6225364305 - 0.4114366382i\)
\(L(1)\) \(\approx\) \(0.6225364305 - 0.4114366382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (0.695 - 0.718i)T \)
3 \( 1 + (-0.946 + 0.323i)T \)
5 \( 1 + (-0.825 - 0.564i)T \)
7 \( 1 + (-0.936 - 0.351i)T \)
11 \( 1 + (0.0466 + 0.998i)T \)
13 \( 1 + (0.974 + 0.224i)T \)
17 \( 1 + (0.958 + 0.286i)T \)
19 \( 1 + (-0.430 - 0.902i)T \)
23 \( 1 + (-0.674 - 0.738i)T \)
29 \( 1 + (-0.0908 - 0.995i)T \)
31 \( 1 + (-0.221 - 0.975i)T \)
37 \( 1 + (-0.508 - 0.861i)T \)
41 \( 1 + (-0.962 + 0.271i)T \)
43 \( 1 + (0.558 - 0.829i)T \)
47 \( 1 + (-0.974 + 0.224i)T \)
53 \( 1 + (0.814 + 0.580i)T \)
59 \( 1 + (-0.999 + 0.0245i)T \)
61 \( 1 + (-0.197 - 0.980i)T \)
67 \( 1 + (-0.362 - 0.931i)T \)
71 \( 1 + (-0.637 + 0.770i)T \)
73 \( 1 + (-0.188 - 0.982i)T \)
79 \( 1 + (-0.699 + 0.714i)T \)
83 \( 1 + (0.159 - 0.987i)T \)
89 \( 1 + (-0.478 + 0.878i)T \)
97 \( 1 + (0.0368 + 0.999i)T \)
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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

−19.80933913762024019679674315408, −18.99292988870775212926648088966, −18.46561746422557078660415376996, −17.89485321473953952815517331416, −16.70818675486890859652891204170, −16.27547762753435715877677133406, −15.92945074776168899745753214383, −15.129959303990223790640218465878, −14.21254366090364260814753438685, −13.51528278357305817708226424523, −12.76737413704779947192947219299, −12.07303190440125697697498338807, −11.62659131160510620363044026429, −10.76990637194917030608720448702, −10.02012900502560591679485737317, −8.65885462013733053098370918528, −8.112516264208784896976036837523, −7.196297243926243808219778506881, −6.62226955126046912975731936677, −5.85464298581735299453496459648, −5.51086854145726204888268240428, −4.28964851062563512800571074013, −3.3786886507666156984384789376, −3.12815813026173932744778310548, −1.45896176954983561695551791525, 0.037505492918069948147297064251, 0.905128660609760188233229585528, 1.95451588869059564812679227957, 3.28003539071409804633851505190, 4.05933850911675479633051843408, 4.353582988015033071529060340422, 5.31898189209789263316218644090, 6.141129478792029903331724396913, 6.75258377064796205102910475752, 7.69029479669954162174713302229, 8.99727953107347334215294563119, 9.62513848189963342185169820615, 10.39713460904495270706300210977, 10.95667073368487514890316531751, 11.876344100749289739471900824746, 12.24561492285665994323907528586, 12.94764256459146660133509124053, 13.49035610447833167511088687953, 14.68756228950109747185668362819, 15.461947381741203774160641721925, 15.845083027075956653555552908327, 16.64461013006323834184846201906, 17.301751849081182015746018811941, 18.40690229175499150354835309317, 18.91984791162526076799656724051

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