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Properties

Label 1-1259-1259.1077-r1-0-0
Degree $1$
Conductor $1259$
Sign $0.869 - 0.493i$
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  + (0.0872 + 0.996i)2-s + (0.0224 − 0.999i)3-s + (−0.984 + 0.173i)4-s + (0.936 − 0.351i)5-s + (0.997 − 0.0648i)6-s + (0.882 + 0.470i)7-s + (−0.259 − 0.965i)8-s + (−0.998 − 0.0449i)9-s + (0.432 + 0.901i)10-s + (−0.0723 − 0.997i)11-s + (0.151 + 0.988i)12-s + (−0.335 + 0.942i)13-s + (−0.391 + 0.920i)14-s + (−0.330 − 0.943i)15-s + (0.939 − 0.342i)16-s + (−0.340 + 0.940i)17-s + ⋯
L(s)  = 1  + (0.0872 + 0.996i)2-s + (0.0224 − 0.999i)3-s + (−0.984 + 0.173i)4-s + (0.936 − 0.351i)5-s + (0.997 − 0.0648i)6-s + (0.882 + 0.470i)7-s + (−0.259 − 0.965i)8-s + (−0.998 − 0.0449i)9-s + (0.432 + 0.901i)10-s + (−0.0723 − 0.997i)11-s + (0.151 + 0.988i)12-s + (−0.335 + 0.942i)13-s + (−0.391 + 0.920i)14-s + (−0.330 − 0.943i)15-s + (0.939 − 0.342i)16-s + (−0.340 + 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ 0.869 - 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.271893862 - 0.5999489350i\)
\(L(\frac12)\) \(\approx\) \(2.271893862 - 0.5999489350i\)
\(L(1)\) \(\approx\) \(1.234364767 + 0.09892618691i\)
\(L(1)\) \(\approx\) \(1.234364767 + 0.09892618691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.0872 + 0.996i)T \)
3 \( 1 + (0.0224 - 0.999i)T \)
5 \( 1 + (0.936 - 0.351i)T \)
7 \( 1 + (0.882 + 0.470i)T \)
11 \( 1 + (-0.0723 - 0.997i)T \)
13 \( 1 + (-0.335 + 0.942i)T \)
17 \( 1 + (-0.340 + 0.940i)T \)
19 \( 1 + (-0.220 - 0.975i)T \)
23 \( 1 + (-0.358 + 0.933i)T \)
29 \( 1 + (0.828 + 0.559i)T \)
31 \( 1 + (-0.921 - 0.389i)T \)
37 \( 1 + (0.966 - 0.256i)T \)
41 \( 1 + (-0.524 - 0.851i)T \)
43 \( 1 + (-0.679 - 0.733i)T \)
47 \( 1 + (0.870 - 0.492i)T \)
53 \( 1 + (0.549 - 0.835i)T \)
59 \( 1 + (0.865 + 0.500i)T \)
61 \( 1 + (0.622 + 0.782i)T \)
67 \( 1 + (-0.811 - 0.584i)T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (0.925 - 0.379i)T \)
79 \( 1 + (0.999 - 0.00998i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (-0.998 + 0.0598i)T \)
97 \( 1 + (0.909 - 0.416i)T \)
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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

−20.83783440491699321519521970078, −20.360003663419087448631483577856, −19.908483409018379279826523934515, −18.437949120741259613865188796168, −17.95464276117430545748815054708, −17.31445232818721973527985699206, −16.579358416158158009366709355518, −15.23169231814744295318823401592, −14.55820176586181422674799962835, −14.14156289750236807536969198477, −13.19814425822148076084428218600, −12.28258226423507958076531719225, −11.3699217068717992262595589945, −10.590594999942947994445374054378, −10.0802993892478608593371861735, −9.58099450799976906469649831586, −8.53389985332025729563479198734, −7.70093627877408174633734992309, −6.26550185003839258771646171716, −5.16959602926470464344027226776, −4.7744005626746758807892810829, −3.849267716421784575326307629055, −2.72273061994605804538116032112, −2.13617213540797061940251448949, −0.88965885221534933731069695816, 0.528986649491440626774359130966, 1.61874122220706608663734769096, 2.441109149166371111822788153427, 3.88322803729826090463510536540, 5.1236785079180176605617157925, 5.61179584213558149751255016137, 6.42700339041521424945426417208, 7.13271323135184396557184384043, 8.20236432736660343367495516901, 8.73788156562216236525864416575, 9.30377792082687051161033910234, 10.68371842976751706068969273251, 11.67284815052447637612251393903, 12.4977046308590899394721302967, 13.42855105675379161824613775043, 13.76383615731595241208941133286, 14.535343874728961112161901630210, 15.27088212330342026568681046217, 16.513626796625842456175880737056, 17.000529285704886668770529154851, 17.80571952862983063488470129190, 18.17547325225169564717680931836, 19.064600483742324243536189341054, 19.79596579462457195655177281949, 21.117640755045937743087242721674

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