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Properties

Label 1-13e2-169.139-r0-0-0
Degree $1$
Conductor $169$
Sign $-0.981 - 0.190i$
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  + (−0.632 + 0.774i)2-s + (−0.996 − 0.0804i)3-s + (−0.200 − 0.979i)4-s + (0.120 + 0.992i)5-s + (0.692 − 0.721i)6-s + (−0.845 + 0.534i)7-s + (0.885 + 0.464i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (0.987 − 0.160i)11-s + (0.120 + 0.992i)12-s + (0.120 − 0.992i)14-s + (−0.0402 − 0.999i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (−0.748 + 0.663i)18-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)2-s + (−0.996 − 0.0804i)3-s + (−0.200 − 0.979i)4-s + (0.120 + 0.992i)5-s + (0.692 − 0.721i)6-s + (−0.845 + 0.534i)7-s + (0.885 + 0.464i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (0.987 − 0.160i)11-s + (0.120 + 0.992i)12-s + (0.120 − 0.992i)14-s + (−0.0402 − 0.999i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (−0.748 + 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.981 - 0.190i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ -0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02840863958 + 0.2948754027i\)
\(L(\frac12)\) \(\approx\) \(0.02840863958 + 0.2948754027i\)
\(L(1)\) \(\approx\) \(0.3678951017 + 0.2769907293i\)
\(L(1)\) \(\approx\) \(0.3678951017 + 0.2769907293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.632 + 0.774i)T \)
3 \( 1 + (-0.996 - 0.0804i)T \)
5 \( 1 + (0.120 + 0.992i)T \)
7 \( 1 + (-0.845 + 0.534i)T \)
11 \( 1 + (0.987 - 0.160i)T \)
17 \( 1 + (-0.845 + 0.534i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.632 + 0.774i)T \)
31 \( 1 + (-0.970 - 0.239i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (-0.996 - 0.0804i)T \)
43 \( 1 + (0.278 - 0.960i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (-0.919 - 0.391i)T \)
61 \( 1 + (-0.0402 + 0.999i)T \)
67 \( 1 + (-0.200 + 0.979i)T \)
71 \( 1 + (0.428 + 0.903i)T \)
73 \( 1 + (-0.354 + 0.935i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.799 + 0.600i)T \)
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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

−27.474842509296133229592274355533, −26.3918931723366466594261363949, −25.246681890550250414899650736213, −24.15425048675486346599347890227, −22.95895636871480837935317496519, −22.12225643310822401562268957407, −21.266709096614734566111363655037, −20.02191386405381674923945347914, −19.51240762428800708121621486470, −18.04853023087530696272988146995, −17.224303916914819818892176552600, −16.589581603409328044253905592960, −15.72135442056901384011234374564, −13.48841728364054206801037923338, −12.80794086478722363779162605000, −11.82360078172520930103174914086, −10.95721495571070588521966925472, −9.641661293320670738200572858768, −9.15286514377023750618590638819, −7.452555317137602313937558676293, −6.32754383459150072261066045326, −4.69666818499755831247786605373, −3.783654618853439422617234845186, −1.72795687921730348819313867581, −0.338235113139928641635494620079, 1.88070330788894836621418030329, 3.96964960156148239483055064400, 5.69709006041037315800644580641, 6.415066230516519338399260784053, 7.064823157957693981431214280621, 8.690670110192562459626937256659, 9.916171386849875741868969322221, 10.6812426923152857713462373229, 11.80503599780172449689730362844, 13.15444806064403458460301610614, 14.54223268825439836786668851782, 15.38105021268785230499324909443, 16.475079956223772563701512164708, 17.17381397066809977193130959814, 18.39143341822491084941662758240, 18.76498267259147468350228232070, 19.89446874577691914384641849787, 21.96949819817459554837120544812, 22.32748888032837586490643333146, 23.27107449374082553059558555655, 24.322246583509956682075169358571, 25.28773504498254991646944495164, 26.16057697694338674527181835031, 27.1739345905652516535626542479, 27.91682162445675625971908926165

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