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Properties

Label 1-1309-1309.930-r1-0-0
Degree $1$
Conductor $1309$
Sign $0.959 - 0.283i$
Analytic cond. $140.671$
Root an. cond. $140.671$
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  + (−0.156 + 0.987i)2-s + (−0.996 − 0.0784i)3-s + (−0.951 − 0.309i)4-s + (0.522 + 0.852i)5-s + (0.233 − 0.972i)6-s + (0.453 − 0.891i)8-s + (0.987 + 0.156i)9-s + (−0.923 + 0.382i)10-s + (0.923 + 0.382i)12-s + (−0.587 − 0.809i)13-s + (−0.453 − 0.891i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.891 − 0.453i)19-s + (−0.233 − 0.972i)20-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.996 − 0.0784i)3-s + (−0.951 − 0.309i)4-s + (0.522 + 0.852i)5-s + (0.233 − 0.972i)6-s + (0.453 − 0.891i)8-s + (0.987 + 0.156i)9-s + (−0.923 + 0.382i)10-s + (0.923 + 0.382i)12-s + (−0.587 − 0.809i)13-s + (−0.453 − 0.891i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.891 − 0.453i)19-s + (−0.233 − 0.972i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $0.959 - 0.283i$
Analytic conductor: \(140.671\)
Root analytic conductor: \(140.671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (1:\ ),\ 0.959 - 0.283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5935192045 - 0.08584911126i\)
\(L(\frac12)\) \(\approx\) \(0.5935192045 - 0.08584911126i\)
\(L(1)\) \(\approx\) \(0.5718275020 + 0.3159084655i\)
\(L(1)\) \(\approx\) \(0.5718275020 + 0.3159084655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.156 + 0.987i)T \)
3 \( 1 + (-0.996 - 0.0784i)T \)
5 \( 1 + (0.522 + 0.852i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (0.649 - 0.760i)T \)
31 \( 1 + (0.233 + 0.972i)T \)
37 \( 1 + (-0.760 - 0.649i)T \)
41 \( 1 + (0.649 + 0.760i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (0.972 + 0.233i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.852 + 0.522i)T \)
73 \( 1 + (0.649 - 0.760i)T \)
79 \( 1 + (-0.852 - 0.522i)T \)
83 \( 1 + (-0.987 + 0.156i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.972 + 0.233i)T \)
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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

−20.91922134901041151132291281677, −20.247611198817727214538382992504, −19.12093152301709488184668312764, −18.6782924280866865512499771037, −17.67743533771199348570171692653, −17.0153975302742326899175478242, −16.73444752054661471020195434337, −15.671658119417490547892459492518, −14.415039851159433740405543430464, −13.65933626406381344690321966789, −12.578424314815936217879255927891, −12.445505639129210797579235466065, −11.56597435881429634229628607096, −10.623983942141480100485264654387, −10.099027198542396536614297056097, −9.19695146545818595127834951720, −8.5867009412313566313530104544, −7.370361351606920038838785458789, −6.26424958815981604913877920592, −5.396156235438043493352535796504, −4.544455443602120539731102290879, −4.10560683061962230447355005463, −2.52579478712451995344053504185, −1.66008298606198907107722375371, −0.76069793933742027437496953363, 0.20223569780266115020643058609, 1.367271877755465262778049645451, 2.731131130959692281048528139330, 4.02874148033457380613495212066, 5.00668047377895972499179374971, 5.70377251158349910176258399064, 6.40146778714816441658793468934, 7.122450550010051847322618430889, 7.76038529480171126844574081063, 8.97428015145021681954199646685, 9.940414199554218423466448254515, 10.43310999241098856192906754625, 11.24272356938487765460141218142, 12.39791232630694069813828785593, 13.13792444623831488575677425696, 13.88659092237719794731908336220, 14.85021604757088410554184817404, 15.40931442599761236091687023004, 16.18190717794801859545211444316, 17.1673818434822035351540900924, 17.66220747750603922351385370628, 17.97609505532085568405712882072, 19.10836002186492494104677007404, 19.47386832521139151865467143218, 21.2091789089895708372810513281

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