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Properties

Label 1-13e2-169.69-r0-0-0
Degree $1$
Conductor $169$
Sign $-0.637 - 0.770i$
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  + (−0.692 − 0.721i)2-s + (0.799 − 0.600i)3-s + (−0.0402 + 0.999i)4-s + (−0.568 − 0.822i)5-s + (−0.987 − 0.160i)6-s + (0.200 + 0.979i)7-s + (0.748 − 0.663i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (−0.278 − 0.960i)11-s + (0.568 + 0.822i)12-s + (0.568 − 0.822i)14-s + (−0.948 − 0.316i)15-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (−0.885 + 0.464i)18-s + ⋯
L(s)  = 1  + (−0.692 − 0.721i)2-s + (0.799 − 0.600i)3-s + (−0.0402 + 0.999i)4-s + (−0.568 − 0.822i)5-s + (−0.987 − 0.160i)6-s + (0.200 + 0.979i)7-s + (0.748 − 0.663i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (−0.278 − 0.960i)11-s + (0.568 + 0.822i)12-s + (0.568 − 0.822i)14-s + (−0.948 − 0.316i)15-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (−0.885 + 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ -0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3766912697 - 0.8002522571i\)
\(L(\frac12)\) \(\approx\) \(0.3766912697 - 0.8002522571i\)
\(L(1)\) \(\approx\) \(0.6804360590 - 0.5485134552i\)
\(L(1)\) \(\approx\) \(0.6804360590 - 0.5485134552i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.692 - 0.721i)T \)
3 \( 1 + (0.799 - 0.600i)T \)
5 \( 1 + (-0.568 - 0.822i)T \)
7 \( 1 + (0.200 + 0.979i)T \)
11 \( 1 + (-0.278 - 0.960i)T \)
17 \( 1 + (-0.200 - 0.979i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.692 + 0.721i)T \)
31 \( 1 + (0.354 + 0.935i)T \)
37 \( 1 + (0.632 - 0.774i)T \)
41 \( 1 + (-0.799 + 0.600i)T \)
43 \( 1 + (-0.632 - 0.774i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (0.996 - 0.0804i)T \)
61 \( 1 + (0.948 - 0.316i)T \)
67 \( 1 + (0.0402 + 0.999i)T \)
71 \( 1 + (0.919 + 0.391i)T \)
73 \( 1 + (0.970 + 0.239i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.120 + 0.992i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.428 - 0.903i)T \)
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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

−27.431202139702547428000929363804, −26.88547994446919770339471020503, −26.04788264212528847513670877733, −25.518381270280996843866335795075, −24.159933161828331817213678821428, −23.24307690967633379546186209980, −22.365998140905239013829446074604, −20.820202657384394276009813442296, −19.90546886009137179512336894499, −19.24661223533340013435298956709, −18.09663960949105725326566714763, −17.04916753557163513111764004104, −15.91203157434652491788673868975, −15.10668817232631063772257914928, −14.41785275421455048923800561004, −13.42390089408859184419616289850, −11.391123219287327996306426362571, −10.21067357207292858837338016776, −9.8476146744400905058484566442, −8.06086439102757026535514257029, −7.72278525702283182072319364996, −6.48300669139876043344071865090, −4.71716855467127151408300959935, −3.66565830968401172932155946260, −1.919346712870830702965551446258, 0.87148231928681507073732471034, 2.38454732458907510214140432589, 3.37132643542602362941025066396, 4.95437099346271017536728306284, 6.88454868727625498697382621875, 8.24588858435499389027286078562, 8.61578517672297986508810918011, 9.580400802798219194594484098712, 11.30806687858213303854615630418, 12.1048499661666762843641154077, 12.951461789567980302518363926085, 13.99727559295822768740458196122, 15.59650720927229480012560620589, 16.33636476224653095668806787182, 17.89000393806149309901147460626, 18.53622202028154277864225150308, 19.489798045059867164331681892489, 20.203717119582512112396028911911, 21.096487323353044202751881502347, 22.01623641223971591839287454531, 23.63560127903051503965050827318, 24.679694702738678455741553793373, 25.19631278062791602927925659393, 26.55722337776834187168570768868, 27.14099609179543446258021311655

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