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Properties

Label 1-1309-1309.13-r0-0-0
Degree $1$
Conductor $1309$
Sign $-0.254 + 0.967i$
Analytic cond. $6.07897$
Root an. cond. $6.07897$
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  + (−0.809 − 0.587i)2-s + (−0.951 − 0.309i)3-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.587 + 0.809i)6-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)9-s − i·10-s − i·12-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.587 + 0.809i)20-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.951 − 0.309i)3-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.587 + 0.809i)6-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)9-s − i·10-s − i·12-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.587 + 0.809i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $-0.254 + 0.967i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ -0.254 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2275713117 + 0.2952857388i\)
\(L(\frac12)\) \(\approx\) \(0.2275713117 + 0.2952857388i\)
\(L(1)\) \(\approx\) \(0.5224920820 + 0.02438756001i\)
\(L(1)\) \(\approx\) \(0.5224920820 + 0.02438756001i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.587 + 0.809i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.587 - 0.809i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−20.7388265347047772860375287244, −19.854974366295540415102839089389, −19.11860516236189376586197426222, −18.137312405905109139629871769950, −17.41738644722790880207629362199, −17.124877061764281991775508175132, −16.310389022476534696174277926777, −15.725997263957632295515381660242, −14.89144516043066525278088123587, −13.90603003372164090428739995488, −13.00896959721366448344563046714, −12.02390940393434912898928663528, −11.33835948752070354134783443218, −10.405303141296185329401550553006, −9.65779037451816245400065865146, −9.18085188649465321374802366496, −8.22555209080523797874962129138, −7.04551264374413339278465358067, −6.55952434629120703058128746470, −5.386906361589213267373351918534, −5.14088536291756573521450953815, −4.09503548633902025130709568266, −2.28701698046923122489746526472, −1.36147810539909054314973637256, −0.23962100945219198103665850023, 1.18599329670268128207037414430, 2.151415331655404718050253866169, 2.94907899541102739891079574090, 4.15025684871766676179489993969, 5.29682220194715361368294138556, 6.26546083326185369366143138527, 6.973155137289739114382073275929, 7.678170067464502186086185872094, 8.677006814734902148673306110753, 9.85673330751726100502713750007, 10.41344755641307162967196229006, 10.782345264564354555378750555978, 11.93291590197068535939318981974, 12.36513860376123350556262524120, 13.21980625252122708217982988191, 14.18462037071552806939201669638, 15.19860844740190536466473008614, 16.204165393265966919013953590343, 16.88241738669800997421357899015, 17.67883469342608398517819370231, 17.96103175560873195798147761735, 18.93660734510502125873573148607, 19.26328453316202623484103976666, 20.45090177066591839408118527, 21.220901373730491554902432635064

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