Properties

Label 1-2557-2557.1034-r0-0-0
Degree 11
Conductor 25572557
Sign 0.151+0.988i0.151 + 0.988i
Analytic cond. 11.874611.8746
Root an. cond. 11.874611.8746
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(2557s/2ΓR(s)L(s)=((0.151+0.988i)Λ(1s)\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}
Λ(s)=(2557s/2ΓR(s)L(s)=((0.151+0.988i)Λ(1s)\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: 11
Conductor: 25572557
Sign: 0.151+0.988i0.151 + 0.988i
Analytic conductor: 11.874611.8746
Root analytic conductor: 11.874611.8746
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2557(1034,)\chi_{2557} (1034, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2557, (0: ), 0.151+0.988i)(1,\ 2557,\ (0:\ ),\ 0.151 + 0.988i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.078205091640.06716394735i-0.07820509164 - 0.06716394735i
L(12)L(\frac12) \approx 0.078205091640.06716394735i-0.07820509164 - 0.06716394735i
L(1)L(1) \approx 0.62253643050.4114366382i0.6225364305 - 0.4114366382i
L(1)L(1) \approx 0.62253643050.4114366382i0.6225364305 - 0.4114366382i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2557 1 1
good2 1+(0.6950.718i)T 1 + (0.695 - 0.718i)T
3 1+(0.946+0.323i)T 1 + (-0.946 + 0.323i)T
5 1+(0.8250.564i)T 1 + (-0.825 - 0.564i)T
7 1+(0.9360.351i)T 1 + (-0.936 - 0.351i)T
11 1+(0.0466+0.998i)T 1 + (0.0466 + 0.998i)T
13 1+(0.974+0.224i)T 1 + (0.974 + 0.224i)T
17 1+(0.958+0.286i)T 1 + (0.958 + 0.286i)T
19 1+(0.4300.902i)T 1 + (-0.430 - 0.902i)T
23 1+(0.6740.738i)T 1 + (-0.674 - 0.738i)T
29 1+(0.09080.995i)T 1 + (-0.0908 - 0.995i)T
31 1+(0.2210.975i)T 1 + (-0.221 - 0.975i)T
37 1+(0.5080.861i)T 1 + (-0.508 - 0.861i)T
41 1+(0.962+0.271i)T 1 + (-0.962 + 0.271i)T
43 1+(0.5580.829i)T 1 + (0.558 - 0.829i)T
47 1+(0.974+0.224i)T 1 + (-0.974 + 0.224i)T
53 1+(0.814+0.580i)T 1 + (0.814 + 0.580i)T
59 1+(0.999+0.0245i)T 1 + (-0.999 + 0.0245i)T
61 1+(0.1970.980i)T 1 + (-0.197 - 0.980i)T
67 1+(0.3620.931i)T 1 + (-0.362 - 0.931i)T
71 1+(0.637+0.770i)T 1 + (-0.637 + 0.770i)T
73 1+(0.1880.982i)T 1 + (-0.188 - 0.982i)T
79 1+(0.699+0.714i)T 1 + (-0.699 + 0.714i)T
83 1+(0.1590.987i)T 1 + (0.159 - 0.987i)T
89 1+(0.478+0.878i)T 1 + (-0.478 + 0.878i)T
97 1+(0.0368+0.999i)T 1 + (0.0368 + 0.999i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line