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Properties

Label 1-13e2-169.14-r0-0-0
Degree $1$
Conductor $169$
Sign $0.656 - 0.754i$
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  + (−0.354 − 0.935i)2-s + (0.568 − 0.822i)3-s + (−0.748 + 0.663i)4-s + (0.120 + 0.992i)5-s + (−0.970 − 0.239i)6-s + (0.885 + 0.464i)7-s + (0.885 + 0.464i)8-s + (−0.354 − 0.935i)9-s + (0.885 − 0.464i)10-s + (−0.354 + 0.935i)11-s + (0.120 + 0.992i)12-s + (0.120 − 0.992i)14-s + (0.885 + 0.464i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.748 + 0.663i)18-s + ⋯
L(s)  = 1  + (−0.354 − 0.935i)2-s + (0.568 − 0.822i)3-s + (−0.748 + 0.663i)4-s + (0.120 + 0.992i)5-s + (−0.970 − 0.239i)6-s + (0.885 + 0.464i)7-s + (0.885 + 0.464i)8-s + (−0.354 − 0.935i)9-s + (0.885 − 0.464i)10-s + (−0.354 + 0.935i)11-s + (0.120 + 0.992i)12-s + (0.120 − 0.992i)14-s + (0.885 + 0.464i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.748 + 0.663i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.656 - 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060387038 - 0.4831800888i\)
\(L(\frac12)\) \(\approx\) \(1.060387038 - 0.4831800888i\)
\(L(1)\) \(\approx\) \(1.003772050 - 0.4115841150i\)
\(L(1)\) \(\approx\) \(1.003772050 - 0.4115841150i\)

Euler product

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

−27.35259844925815963691846602388, −26.90152610366859922792631682010, −25.801577473056825454585263597140, −24.83506605934721985354951204154, −24.16891801470145516768843681998, −23.16000654700438467824202727184, −21.78657920066413410650788422100, −20.85556893580164148954357084852, −20.04068033384231586469104331478, −18.84070261863926826411576536683, −17.62048837942496522980286332128, −16.46915225067871575146473430248, −16.211523292358731956224616394846, −14.84400789866380799633936300555, −14.03835710973215130630285005796, −13.19372309533895504533991344533, −11.29453803915482992191881524845, −10.13238368706128693081532133808, −9.08256260501708840860057488435, −8.322620829864045566300699162624, −7.4285890689065955572590939523, −5.37169682637185400285551842867, −4.99543606242914925987031829868, −3.549783501995109746242995184837, −1.2678198608080089069143753144, 1.60673840763008650271934688312, 2.50019434622261740633855896239, 3.627728058392439829597974988730, 5.399835175515155323063665511542, 7.222741595963736645430613029695, 7.87987302095536574247921859254, 9.13441976422569567412447949941, 10.22299311899985872062540564888, 11.42780834517711420137539734442, 12.22781779001867313606718750866, 13.34846469433771816022788090283, 14.38068489143332396396723937590, 15.12712872408817494729208531687, 17.22766870865719657640536093378, 18.08625167129548567822234111822, 18.58841518240548845079239442969, 19.505846382277456602260195895832, 20.635838720497601586902867574637, 21.30816223633610409956953912159, 22.59545596926912378503983018404, 23.39356306602591303878300253386, 24.77209276128608087013791727040, 25.755555386996995692036542999496, 26.443029974003392987639838806194, 27.50897815248695758376601020253

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