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Properties

Label 1-13e2-169.3-r0-0-0
Degree $1$
Conductor $169$
Sign $0.348 - 0.937i$
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  + (0.278 − 0.960i)2-s + (−0.919 − 0.391i)3-s + (−0.845 − 0.534i)4-s + (0.568 + 0.822i)5-s + (−0.632 + 0.774i)6-s + (0.948 + 0.316i)7-s + (−0.748 + 0.663i)8-s + (0.692 + 0.721i)9-s + (0.948 − 0.316i)10-s + (0.692 − 0.721i)11-s + (0.568 + 0.822i)12-s + (0.568 − 0.822i)14-s + (−0.200 − 0.979i)15-s + (0.428 + 0.903i)16-s + (0.948 + 0.316i)17-s + (0.885 − 0.464i)18-s + ⋯
L(s)  = 1  + (0.278 − 0.960i)2-s + (−0.919 − 0.391i)3-s + (−0.845 − 0.534i)4-s + (0.568 + 0.822i)5-s + (−0.632 + 0.774i)6-s + (0.948 + 0.316i)7-s + (−0.748 + 0.663i)8-s + (0.692 + 0.721i)9-s + (0.948 − 0.316i)10-s + (0.692 − 0.721i)11-s + (0.568 + 0.822i)12-s + (0.568 − 0.822i)14-s + (−0.200 − 0.979i)15-s + (0.428 + 0.903i)16-s + (0.948 + 0.316i)17-s + (0.885 − 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9036120163 - 0.6278606489i\)
\(L(\frac12)\) \(\approx\) \(0.9036120163 - 0.6278606489i\)
\(L(1)\) \(\approx\) \(0.9260666194 - 0.4756663992i\)
\(L(1)\) \(\approx\) \(0.9260666194 - 0.4756663992i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.278 - 0.960i)T \)
3 \( 1 + (-0.919 - 0.391i)T \)
5 \( 1 + (0.568 + 0.822i)T \)
7 \( 1 + (0.948 + 0.316i)T \)
11 \( 1 + (0.692 - 0.721i)T \)
17 \( 1 + (0.948 + 0.316i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.278 - 0.960i)T \)
31 \( 1 + (-0.354 - 0.935i)T \)
37 \( 1 + (0.987 + 0.160i)T \)
41 \( 1 + (-0.919 - 0.391i)T \)
43 \( 1 + (0.987 - 0.160i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
53 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (0.428 - 0.903i)T \)
61 \( 1 + (-0.200 + 0.979i)T \)
67 \( 1 + (-0.845 + 0.534i)T \)
71 \( 1 + (0.799 - 0.600i)T \)
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 \( 1 + (-0.996 - 0.0804i)T \)
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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

−27.56162551868423572179391828650, −27.06583776861530456273527438694, −25.57294076620627848620343812343, −24.8148302461315259690613528424, −23.82825315501877620577436033713, −23.1895727931963276521599193364, −22.07231340769397871183923480089, −21.22730456259167297590008352671, −20.39644501977046014753171398487, −18.347606344902445644046077227445, −17.619474736170441695171584984622, −16.79599986322833587034360824334, −16.269793619802632029602468616055, −14.82772694716753874261844526852, −14.1199537783287653441197397770, −12.61185524248444187267263103732, −12.04257361950599756435714681280, −10.38462856943549264709248760993, −9.35777720732713539330966031206, −8.1727421088690030206616664876, −6.86490830212246932098292122558, −5.7262230896612307599207693511, −4.85457996656843720721912493882, −4.05718352803946889179164871183, −1.30501235460012084248814692268, 1.29952044805434409124111529482, 2.45512511296859713692982607522, 4.08950791307026141221918257888, 5.48891766041543261042940794935, 6.189066139432390467050403722158, 7.81632047368182663443388924783, 9.36854662723937592711439463766, 10.56102262403440258681297765496, 11.31726102645580594602497375815, 11.99482311274735564208905404938, 13.34555108191700156702393591710, 14.150594356872580636804545663240, 15.21108147260162837376279436310, 17.08781909855781872458015739394, 17.7075316273348977000566186940, 18.69082443866990150303330983420, 19.29572999011618132242948108162, 20.92225478107181512224095421361, 21.815119917150023327398335962856, 22.18883362987706038227147076643, 23.46005775224888621122151883150, 24.12089694465931707394957757125, 25.37240950424633755663435609776, 26.88577597571295505159150461971, 27.6469768324269734966025826145

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