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Properties

Label 1-2653-2653.2393-r1-0-0
Degree $1$
Conductor $2653$
Sign $0.211 + 0.977i$
Analytic cond. $285.104$
Root an. cond. $285.104$
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  + (−0.222 − 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 + 0.781i)5-s + (0.900 + 0.433i)6-s + (0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (0.900 + 0.433i)10-s + 11-s + (0.222 − 0.974i)12-s + (0.900 + 0.433i)13-s + (−0.222 − 0.974i)15-s + (0.623 − 0.781i)16-s + (−0.623 − 0.781i)17-s + (−0.900 + 0.433i)18-s + (0.222 + 0.974i)19-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 + 0.781i)5-s + (0.900 + 0.433i)6-s + (0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (0.900 + 0.433i)10-s + 11-s + (0.222 − 0.974i)12-s + (0.900 + 0.433i)13-s + (−0.222 − 0.974i)15-s + (0.623 − 0.781i)16-s + (−0.623 − 0.781i)17-s + (−0.900 + 0.433i)18-s + (0.222 + 0.974i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2653 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2653 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2653\)    =    \(7 \cdot 379\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(285.104\)
Root analytic conductor: \(285.104\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2653} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2653,\ (1:\ ),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056212346 + 0.8523198540i\)
\(L(\frac12)\) \(\approx\) \(1.056212346 + 0.8523198540i\)
\(L(1)\) \(\approx\) \(0.7482194804 + 0.06336004477i\)
\(L(1)\) \(\approx\) \(0.7482194804 + 0.06336004477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
379 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + (0.222 + 0.974i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.900 - 0.433i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.222 - 0.974i)T \)
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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

−18.89703211483597754671022482957, −18.09276302340062208351478259219, −17.43079263802755888296036730679, −16.93903648229560572063007275779, −16.325554304591438722277237112558, −15.51802760479587479700805342231, −15.03515388507012268114342894449, −13.79271704954393060637166142143, −13.4372831156027810726323204746, −12.633323299637073018418109420930, −11.990111485012152808689687727377, −11.11266390546948932409702301740, −10.422560622153737046710023883880, −9.17620663149234444901445736916, −8.59574710793445328741477556480, −8.09496390469261565295191852764, −7.169465811445720832278916766480, −6.50664145616652088806439572092, −5.96497524971151059297848630419, −4.94167812727591605267506237101, −4.44854434385717927522881292560, −3.456258175783743711120288901189, −1.86713277338734629532653619261, −0.88343901944263773665072385952, −0.47058466509508017608554541899, 0.78215404336463724435741743292, 1.60692362625804705519424264117, 3.115441168972381659819457533119, 3.35985631001101120060789820513, 4.34597849180948207087099863041, 4.78289298032067035502631594773, 6.10936805227774113354282457481, 6.67433250788416602283712763161, 7.792082852208126932768119522594, 8.66565876225948661751623249056, 9.42459282632338580644799499134, 9.98783330876551613099752239077, 10.880593215891941643386906303337, 11.39148419387136975540288771363, 11.747632087019901165339988690161, 12.52092464179209397429096661017, 13.69969490418547696110396165304, 14.22710748148666155163965847255, 15.0850420059314861717354151385, 15.82074370830079060864390458893, 16.61578632461563497170537806587, 17.207112385075853728671336616652, 18.17407226559884713540310599831, 18.469029148954815818948765833089, 19.3771063916397137338934360493

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