Properties

Label 1-13e2-169.160-r0-0-0
Degree $1$
Conductor $169$
Sign $0.290 + 0.956i$
Analytic cond. $0.784832$
Root an. cond. $0.784832$
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  + (0.845 + 0.534i)2-s + (0.692 + 0.721i)3-s + (0.428 + 0.903i)4-s + (0.354 − 0.935i)5-s + (0.200 + 0.979i)6-s + (−0.799 − 0.600i)7-s + (−0.120 + 0.992i)8-s + (−0.0402 + 0.999i)9-s + (0.799 − 0.600i)10-s + (0.0402 + 0.999i)11-s + (−0.354 + 0.935i)12-s + (−0.354 − 0.935i)14-s + (0.919 − 0.391i)15-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.568 + 0.822i)18-s + ⋯
L(s)  = 1  + (0.845 + 0.534i)2-s + (0.692 + 0.721i)3-s + (0.428 + 0.903i)4-s + (0.354 − 0.935i)5-s + (0.200 + 0.979i)6-s + (−0.799 − 0.600i)7-s + (−0.120 + 0.992i)8-s + (−0.0402 + 0.999i)9-s + (0.799 − 0.600i)10-s + (0.0402 + 0.999i)11-s + (−0.354 + 0.935i)12-s + (−0.354 − 0.935i)14-s + (0.919 − 0.391i)15-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.568 + 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.717262450 + 1.273866369i\)
\(L(\frac12)\) \(\approx\) \(1.717262450 + 1.273866369i\)
\(L(1)\) \(\approx\) \(1.693263514 + 0.8350027949i\)
\(L(1)\) \(\approx\) \(1.693263514 + 0.8350027949i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.845 + 0.534i)T \)
3 \( 1 + (0.692 + 0.721i)T \)
5 \( 1 + (0.354 - 0.935i)T \)
7 \( 1 + (-0.799 - 0.600i)T \)
11 \( 1 + (0.0402 + 0.999i)T \)
17 \( 1 + (0.799 + 0.600i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.845 - 0.534i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (-0.948 - 0.316i)T \)
41 \( 1 + (-0.692 - 0.721i)T \)
43 \( 1 + (0.948 - 0.316i)T \)
47 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (0.120 - 0.992i)T \)
59 \( 1 + (0.632 + 0.774i)T \)
61 \( 1 + (-0.919 - 0.391i)T \)
67 \( 1 + (-0.428 + 0.903i)T \)
71 \( 1 + (-0.278 + 0.960i)T \)
73 \( 1 + (-0.885 - 0.464i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.970 + 0.239i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.987 - 0.160i)T \)
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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

−27.41951658701568082121246703909, −26.15588235873603755670139636300, −25.28506085429831905113087088304, −24.54968070744345552308820805830, −23.37663486521918170214331550581, −22.51713166180901509704442739304, −21.61636222810890059196515274349, −20.68746274209645630691406984228, −19.38882135400450675866582851261, −18.91162852084429561591406778856, −18.12463977618927037792412966239, −16.1852648548425959390966015771, −15.10741253742717083622904249938, −14.09476479056091429329757380916, −13.60153126007216149039123901769, −12.4019261212027593700356873643, −11.55444166543193661782084875854, −10.15029194926797256631498135496, −9.23376458364498399859879343755, −7.560319840546068570797830463688, −6.37254915959889920688634531463, −5.66853319029026190620794776214, −3.34730240792761047756801093752, −3.062775044679363528035913759321, −1.61644359769028694400699395833, 2.24122475844150648227705001244, 3.69547026413505047248819543758, 4.52843483211711105294703807111, 5.62477683608952066295586321847, 7.09505984876113751209712547203, 8.2233731072300416917266246344, 9.38638399306535818412555286639, 10.34142661366753623510503554397, 12.08801684427793103636369619229, 13.08745195727446780472088188330, 13.78516678740244742595804261447, 14.9087708579732938085299247172, 15.85102021271675207028343252893, 16.6420176042399046800287047667, 17.43506349244995973512109486277, 19.433648809417412094851169019820, 20.44499864937836181764657436492, 20.84529281905205665373251164421, 22.07769526392504891566817727189, 22.81875291682317495855120937790, 24.0366832271113470991018723473, 24.920600396280932325711366080058, 25.93587310769643018373646640814, 26.23709502677684474495893506179, 27.78297320719049813550756193349

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