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Properties

Label 1-13e2-169.135-r1-0-0
Degree $1$
Conductor $169$
Sign $-0.754 - 0.656i$
Analytic cond. $18.1615$
Root an. cond. $18.1615$
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  + (−0.935 + 0.354i)2-s + (0.568 − 0.822i)3-s + (0.748 − 0.663i)4-s + (0.992 − 0.120i)5-s + (−0.239 + 0.970i)6-s + (−0.464 + 0.885i)7-s + (−0.464 + 0.885i)8-s + (−0.354 − 0.935i)9-s + (−0.885 + 0.464i)10-s + (−0.935 − 0.354i)11-s + (−0.120 − 0.992i)12-s + (0.120 − 0.992i)14-s + (0.464 − 0.885i)15-s + (0.120 − 0.992i)16-s + (−0.885 − 0.464i)17-s + (0.663 + 0.748i)18-s + ⋯
L(s)  = 1  + (−0.935 + 0.354i)2-s + (0.568 − 0.822i)3-s + (0.748 − 0.663i)4-s + (0.992 − 0.120i)5-s + (−0.239 + 0.970i)6-s + (−0.464 + 0.885i)7-s + (−0.464 + 0.885i)8-s + (−0.354 − 0.935i)9-s + (−0.885 + 0.464i)10-s + (−0.935 − 0.354i)11-s + (−0.120 − 0.992i)12-s + (0.120 − 0.992i)14-s + (0.464 − 0.885i)15-s + (0.120 − 0.992i)16-s + (−0.885 − 0.464i)17-s + (0.663 + 0.748i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.754 - 0.656i$
Analytic conductor: \(18.1615\)
Root analytic conductor: \(18.1615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (1:\ ),\ -0.754 - 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2888575913 - 0.7724631222i\)
\(L(\frac12)\) \(\approx\) \(0.2888575913 - 0.7724631222i\)
\(L(1)\) \(\approx\) \(0.7390307214 - 0.2295793435i\)
\(L(1)\) \(\approx\) \(0.7390307214 - 0.2295793435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.935 + 0.354i)T \)
3 \( 1 + (0.568 - 0.822i)T \)
5 \( 1 + (0.992 - 0.120i)T \)
7 \( 1 + (-0.464 + 0.885i)T \)
11 \( 1 + (-0.935 - 0.354i)T \)
17 \( 1 + (-0.885 - 0.464i)T \)
19 \( 1 - iT \)
23 \( 1 - T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.239 + 0.970i)T \)
37 \( 1 + (0.239 - 0.970i)T \)
41 \( 1 + (-0.822 - 0.568i)T \)
43 \( 1 + (0.970 - 0.239i)T \)
47 \( 1 + (0.663 - 0.748i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (-0.992 + 0.120i)T \)
61 \( 1 + (0.885 - 0.464i)T \)
67 \( 1 + (-0.663 + 0.748i)T \)
71 \( 1 + (-0.822 - 0.568i)T \)
73 \( 1 + (-0.935 - 0.354i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (0.822 - 0.568i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.992 - 0.120i)T \)
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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

−27.61196575101962881630734836624, −26.57271054276788896636189485847, −25.98040671658733348164176956610, −25.45473468591970997987188737725, −24.137554506544718513843177645517, −22.46534701050646459500719861560, −21.65443602620907909250199007935, −20.56059608591202869008399368358, −20.23859581957437286204248823739, −18.95128681377561087925058093911, −17.89628434989402741115753827471, −16.87560338586872411638113479827, −16.142965402613900797776079772716, −14.97467335020067924720861516743, −13.676589045045952117365322105520, −12.7996535942551527469070016487, −10.97155323420841803236645451184, −10.19626775467083903078725233940, −9.68795342772833311178314115163, −8.4723265808727137492070706199, −7.37079358295993110380190851656, −5.961941958084433022534705583367, −4.213670653206209255195656178473, −2.93234347964565689191792492097, −1.80075248287229001445820315949, 0.33895567909208820320329273585, 2.07063375656193288078158966755, 2.690055995421994243078897083522, 5.471096337417687685936097981964, 6.30891763530763542995733023616, 7.38663714175775161874911291166, 8.717543654666056952957443889956, 9.197838813382527147789535290858, 10.43249387292009097281231999100, 11.84879993122725860451089501968, 13.08909067716207612003378470144, 13.949820313558767069868526434975, 15.23011882710829078266670803855, 16.085745735807598772311366270404, 17.552660107162326055197901247043, 18.1164779418363186974915589894, 18.89695113401450290015349820294, 19.90642957186349845992672396320, 20.86643528992382836820689870662, 21.98653478440928513074351399668, 23.580173820161329786722506801501, 24.53635227125227121289495361397, 25.06933467383269264639372331444, 26.031829482925774333366244567235, 26.48511592993129786449250235817

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