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Properties

Label 1-5265-5265.1168-r0-0-0
Degree $1$
Conductor $5265$
Sign $0.998 + 0.0474i$
Analytic cond. $24.4505$
Root an. cond. $24.4505$
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  + (0.973 − 0.230i)2-s + (0.893 − 0.448i)4-s + (0.0581 + 0.998i)7-s + (0.766 − 0.642i)8-s + (−0.116 + 0.993i)11-s + (0.286 + 0.957i)14-s + (0.597 − 0.802i)16-s + (−0.342 − 0.939i)17-s + (−0.642 − 0.766i)19-s + (0.116 + 0.993i)22-s + (0.998 + 0.0581i)23-s + (0.5 + 0.866i)28-s + (0.686 + 0.727i)29-s + (0.549 − 0.835i)31-s + (0.396 − 0.918i)32-s + ⋯
L(s)  = 1  + (0.973 − 0.230i)2-s + (0.893 − 0.448i)4-s + (0.0581 + 0.998i)7-s + (0.766 − 0.642i)8-s + (−0.116 + 0.993i)11-s + (0.286 + 0.957i)14-s + (0.597 − 0.802i)16-s + (−0.342 − 0.939i)17-s + (−0.642 − 0.766i)19-s + (0.116 + 0.993i)22-s + (0.998 + 0.0581i)23-s + (0.5 + 0.866i)28-s + (0.686 + 0.727i)29-s + (0.549 − 0.835i)31-s + (0.396 − 0.918i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5265\)    =    \(3^{4} \cdot 5 \cdot 13\)
Sign: $0.998 + 0.0474i$
Analytic conductor: \(24.4505\)
Root analytic conductor: \(24.4505\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5265} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5265,\ (0:\ ),\ 0.998 + 0.0474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.778610851 + 0.08961382372i\)
\(L(\frac12)\) \(\approx\) \(3.778610851 + 0.08961382372i\)
\(L(1)\) \(\approx\) \(2.026958179 - 0.08833966854i\)
\(L(1)\) \(\approx\) \(2.026958179 - 0.08833966854i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.973 - 0.230i)T \)
7 \( 1 + (0.0581 + 0.998i)T \)
11 \( 1 + (-0.116 + 0.993i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.998 + 0.0581i)T \)
29 \( 1 + (0.686 + 0.727i)T \)
31 \( 1 + (0.549 - 0.835i)T \)
37 \( 1 + (-0.173 + 0.984i)T \)
41 \( 1 + (0.957 - 0.286i)T \)
43 \( 1 + (-0.116 + 0.993i)T \)
47 \( 1 + (0.835 - 0.549i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.116 - 0.993i)T \)
61 \( 1 + (-0.835 + 0.549i)T \)
67 \( 1 + (0.973 + 0.230i)T \)
71 \( 1 + (0.984 + 0.173i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (0.396 + 0.918i)T \)
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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

−17.61334349535279773153978103926, −17.0973552697774183006117029346, −16.628912807930974444839353505117, −15.85660103485054741964377629320, −15.30477894510834599152014977951, −14.42930747229618591863417140940, −13.98761430622493030444703614531, −13.4278922858670252049196945273, −12.680454631419450960855199604184, −12.20959975193819895705970529350, −11.08309809221273559298169694212, −10.83414868215051270512517185549, −10.22474019430037992696147019088, −9.01687589679474568830942839995, −8.21600658345091843962898247919, −7.72128120152962985707375693472, −6.825411426780181060610095075244, −6.27066782802968796451269000874, −5.63024435798501594126587252411, −4.6997985692785470246146306378, −4.08272514254365809154722650677, −3.48362995646156946904229463673, −2.69726299665429552326810040032, −1.73582765480623818232806349226, −0.81193121971917036480799263034, 0.91069745060428248955459405212, 2.00522963463588560808808924277, 2.55641812433679563618327975248, 3.13751649620595057625207057140, 4.27158032931922748190406800868, 4.9062701652968121697083672351, 5.26981212941071237414256898998, 6.3859676758917035500601812798, 6.75441510571552534974230090255, 7.59134647530144476729313013620, 8.48010741831400619590966433635, 9.40460525103155803893278440676, 9.82611193810592917709287645081, 10.972876555605707431640835843154, 11.2534128000952182672022992242, 12.21192939053515565552495115770, 12.52393357880633976660267691111, 13.25802871559152150120669557238, 13.92880782990160804789094359936, 14.731576132098593043803016721732, 15.32658675757345617711969906200, 15.583474428368295838564105684292, 16.45631744225145911623077446528, 17.294116376326873076220116065030, 18.017033172813568793289692773337

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