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Properties

Label 1-13e2-169.3-r0-0-0
Degree 11
Conductor 169169
Sign 0.348−0.937i0.348 - 0.937i
Analytic cond. 0.7848320.784832
Root an. cond. 0.7848320.784832
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(169s/2 ΓR(s) L(s)=((0.348−0.937i) Λ‾(1−s)\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}
Λ(s)=(169s/2 ΓR(s) L(s)=((0.348−0.937i) Λ‾(1−s)\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: 11
Conductor: 169169    =    13213^{2}
Sign: 0.348−0.937i0.348 - 0.937i
Analytic conductor: 0.7848320.784832
Root analytic conductor: 0.7848320.784832
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ169(3,⋅)\chi_{169} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 169, (0: ), 0.348−0.937i)(1,\ 169,\ (0:\ ),\ 0.348 - 0.937i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.9036120163−0.6278606489i0.9036120163 - 0.6278606489i
L(12)L(\frac12) ≈\approx 0.9036120163−0.6278606489i0.9036120163 - 0.6278606489i
L(1)L(1) ≈\approx 0.9260666194−0.4756663992i0.9260666194 - 0.4756663992i
L(1)L(1) ≈\approx 0.9260666194−0.4756663992i0.9260666194 - 0.4756663992i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad13 1 1
good2 1+(0.278−0.960i)T 1 + (0.278 - 0.960i)T
3 1+(−0.919−0.391i)T 1 + (-0.919 - 0.391i)T
5 1+(0.568+0.822i)T 1 + (0.568 + 0.822i)T
7 1+(0.948+0.316i)T 1 + (0.948 + 0.316i)T
11 1+(0.692−0.721i)T 1 + (0.692 - 0.721i)T
17 1+(0.948+0.316i)T 1 + (0.948 + 0.316i)T
19 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
23 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
29 1+(0.278−0.960i)T 1 + (0.278 - 0.960i)T
31 1+(−0.354−0.935i)T 1 + (-0.354 - 0.935i)T
37 1+(0.987+0.160i)T 1 + (0.987 + 0.160i)T
41 1+(−0.919−0.391i)T 1 + (-0.919 - 0.391i)T
43 1+(0.987−0.160i)T 1 + (0.987 - 0.160i)T
47 1+(0.885+0.464i)T 1 + (0.885 + 0.464i)T
53 1+(−0.748+0.663i)T 1 + (-0.748 + 0.663i)T
59 1+(0.428−0.903i)T 1 + (0.428 - 0.903i)T
61 1+(−0.200+0.979i)T 1 + (-0.200 + 0.979i)T
67 1+(−0.845+0.534i)T 1 + (-0.845 + 0.534i)T
71 1+(0.799−0.600i)T 1 + (0.799 - 0.600i)T
73 1+(−0.970−0.239i)T 1 + (-0.970 - 0.239i)T
79 1+(0.885+0.464i)T 1 + (0.885 + 0.464i)T
83 1+(0.120−0.992i)T 1 + (0.120 - 0.992i)T
89 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
97 1+(−0.996−0.0804i)T 1 + (-0.996 - 0.0804i)T
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   L(s)=∏p (1−αp p−s)−1L(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 ZZ-function along the critical line