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Properties

Label 1-2624-2624.1357-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.701 + 0.712i$
Analytic cond. $12.1858$
Root an. cond. $12.1858$
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  + (−0.382 + 0.923i)3-s + (−0.972 − 0.233i)5-s + (0.891 + 0.453i)7-s + (−0.707 − 0.707i)9-s + (−0.852 − 0.522i)11-s + (−0.649 + 0.760i)13-s + (0.587 − 0.809i)15-s + (0.587 + 0.809i)17-s + (−0.0784 + 0.996i)19-s + (−0.760 + 0.649i)21-s + (0.891 − 0.453i)23-s + (0.891 + 0.453i)25-s + (0.923 − 0.382i)27-s + (−0.852 + 0.522i)29-s + (0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (−0.972 − 0.233i)5-s + (0.891 + 0.453i)7-s + (−0.707 − 0.707i)9-s + (−0.852 − 0.522i)11-s + (−0.649 + 0.760i)13-s + (0.587 − 0.809i)15-s + (0.587 + 0.809i)17-s + (−0.0784 + 0.996i)19-s + (−0.760 + 0.649i)21-s + (0.891 − 0.453i)23-s + (0.891 + 0.453i)25-s + (0.923 − 0.382i)27-s + (−0.852 + 0.522i)29-s + (0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.701 + 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3545465871 + 0.8464512178i\)
\(L(\frac12)\) \(\approx\) \(0.3545465871 + 0.8464512178i\)
\(L(1)\) \(\approx\) \(0.7255546010 + 0.3226714965i\)
\(L(1)\) \(\approx\) \(0.7255546010 + 0.3226714965i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (-0.972 - 0.233i)T \)
7 \( 1 + (0.891 + 0.453i)T \)
11 \( 1 + (-0.852 - 0.522i)T \)
13 \( 1 + (-0.649 + 0.760i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.0784 + 0.996i)T \)
23 \( 1 + (0.891 - 0.453i)T \)
29 \( 1 + (-0.852 + 0.522i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.972 + 0.233i)T \)
43 \( 1 + (0.996 - 0.0784i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.233 - 0.972i)T \)
59 \( 1 + (-0.0784 - 0.996i)T \)
61 \( 1 + (0.996 + 0.0784i)T \)
67 \( 1 + (0.852 - 0.522i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−19.09749053001253743392333055420, −18.27945156539496000229859720876, −17.76419808705821214772333125347, −17.17438454655817451699231356820, −16.32593241680169004652636103545, −15.42784821649069442182609842397, −14.87353415185541434905155183556, −14.0835024141540988141964180183, −13.22526946310178426380775202215, −12.64945413075602737279274397039, −11.85167147161146307095315382160, −11.24627833762280703399330675992, −10.74548387063176916855477425752, −9.793635717374258622051325573013, −8.59164356071597543006635202176, −7.81618919989067756152936839128, −7.40618018691840241011261098187, −6.98835629700979535818315906288, −5.65783536546611082738008304535, −4.97937116538608679360106805956, −4.377033028578147953655019568202, −2.965974867171387599114157523535, −2.51613111853049539426759232737, −1.20535839225574424012246097781, −0.404820459506214425786961556277, 0.96171694338565972703600583815, 2.25377938366543996454347999791, 3.2737119583100361771812297810, 4.04208458223254795842517079322, 4.7721617540042348673336305555, 5.35182466000526488355326770538, 6.154048173574842043178435957011, 7.31785417696647292745145442085, 8.19991431948334502528536315153, 8.53128376094330322498749855742, 9.54100013733895290926768965990, 10.31910376132838940998043913395, 11.199596736818317576858563439805, 11.46340001686301878338742653928, 12.33261571651767005210993603964, 12.95086050671864373396765445918, 14.460649556365709785126506239819, 14.60228021019710082880613806925, 15.403192579617706721038213463221, 16.071641715201695692096645408841, 16.78423218095916334104862848076, 17.166797868711776597884934482767, 18.35319027926612273528732409055, 18.8433398471733803577071407216, 19.58940520794640531212946304126

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