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Properties

Label 1-1259-1259.1055-r1-0-0
Degree $1$
Conductor $1259$
Sign $-0.316 + 0.948i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (−0.745 + 0.666i)2-s + (0.316 − 0.948i)3-s + (0.112 − 0.993i)4-s + (0.427 + 0.903i)5-s + (0.395 + 0.918i)6-s + (−0.950 + 0.309i)7-s + (0.578 + 0.815i)8-s + (−0.799 − 0.600i)9-s + (−0.921 − 0.389i)10-s + (0.870 − 0.492i)11-s + (−0.907 − 0.420i)12-s + (−0.326 − 0.945i)13-s + (0.502 − 0.864i)14-s + (0.992 − 0.119i)15-s + (−0.974 − 0.222i)16-s + (−0.707 + 0.706i)17-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)2-s + (0.316 − 0.948i)3-s + (0.112 − 0.993i)4-s + (0.427 + 0.903i)5-s + (0.395 + 0.918i)6-s + (−0.950 + 0.309i)7-s + (0.578 + 0.815i)8-s + (−0.799 − 0.600i)9-s + (−0.921 − 0.389i)10-s + (0.870 − 0.492i)11-s + (−0.907 − 0.420i)12-s + (−0.326 − 0.945i)13-s + (0.502 − 0.864i)14-s + (0.992 − 0.119i)15-s + (−0.974 − 0.222i)16-s + (−0.707 + 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.316 + 0.948i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ -0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3247531138 + 0.4509195259i\)
\(L(\frac12)\) \(\approx\) \(0.3247531138 + 0.4509195259i\)
\(L(1)\) \(\approx\) \(0.6925904070 + 0.03607131433i\)
\(L(1)\) \(\approx\) \(0.6925904070 + 0.03607131433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.745 + 0.666i)T \)
3 \( 1 + (0.316 - 0.948i)T \)
5 \( 1 + (0.427 + 0.903i)T \)
7 \( 1 + (-0.950 + 0.309i)T \)
11 \( 1 + (0.870 - 0.492i)T \)
13 \( 1 + (-0.326 - 0.945i)T \)
17 \( 1 + (-0.707 + 0.706i)T \)
19 \( 1 + (-0.887 - 0.461i)T \)
23 \( 1 + (0.852 + 0.522i)T \)
29 \( 1 + (-0.372 - 0.927i)T \)
31 \( 1 + (-0.0324 - 0.999i)T \)
37 \( 1 + (-0.0574 - 0.998i)T \)
41 \( 1 + (0.450 - 0.892i)T \)
43 \( 1 + (-0.728 + 0.684i)T \)
47 \( 1 + (0.537 + 0.843i)T \)
53 \( 1 + (-0.0374 - 0.999i)T \)
59 \( 1 + (0.330 + 0.943i)T \)
61 \( 1 + (0.949 + 0.314i)T \)
67 \( 1 + (-0.0474 + 0.998i)T \)
71 \( 1 + (-0.844 - 0.534i)T \)
73 \( 1 + (0.765 + 0.643i)T \)
79 \( 1 + (-0.618 - 0.785i)T \)
83 \( 1 + (-0.602 + 0.798i)T \)
89 \( 1 + (-0.653 + 0.757i)T \)
97 \( 1 + (-0.386 - 0.922i)T \)
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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

−20.3914065079071896312234747382, −20.05444512330098354566027987908, −19.38578694798379229461635181430, −18.51673313235178446281959058177, −17.24976922837908785002317504423, −16.854488111133843427419981683563, −16.37358674514010862655833035702, −15.55397220582099830483377656506, −14.40902897617859834732337886950, −13.56635612036455129492999595133, −12.75090172688043026665059470807, −12.0223539005062208817330999265, −11.10270885762660293036174494782, −10.165195001980291080011321963000, −9.62477068064867998633994378551, −8.93664876280638448254830802878, −8.598643990709387849309307395804, −7.130539768215507622272342351721, −6.43182042411660194937298225528, −4.88733811248583161076302695340, −4.30311125494484379147350119645, −3.44900206391811803821321354485, −2.41533970436368826637700634446, −1.4824936826893223698042486769, −0.16925706142317709463043137169, 0.78951580915326048237675799944, 2.076394735589923537503083702, 2.70783840483786561973795332617, 3.85266679112263581969961483067, 5.71116250655824948060573003331, 6.079325898534013179841717873839, 6.83682337854704831845722097847, 7.41971022420719771433606295676, 8.47910225905153411555149299185, 9.14918431054635180055439364138, 9.89203975946485842455077788636, 10.8895502032288223109010287718, 11.56047049220788116229867659615, 12.93066235330689061967542625038, 13.33480228789244395137357315687, 14.36082759355805769623588199508, 15.00363472769821864171511576561, 15.53144834803266383557201707011, 16.85378377751307490867018440782, 17.40430496450219792781915812036, 17.96438597055145334247460526953, 18.95974236572186699385908058033, 19.33696768404911729684354284079, 19.72142765820783895044230860977, 20.96963490876993336484105703363

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