Properties

Label 1-13-13.4-r0-0-0
Degree 11
Conductor 1313
Sign 0.859+0.511i0.859 + 0.511i
Analytic cond. 0.06037170.0603717
Root an. cond. 0.06037170.0603717
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + ⋯

Functional equation

Λ(s)=(13s/2ΓR(s)L(s)=((0.859+0.511i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(13s/2ΓR(s)L(s)=((0.859+0.511i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 1313
Sign: 0.859+0.511i0.859 + 0.511i
Analytic conductor: 0.06037170.0603717
Root analytic conductor: 0.06037170.0603717
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ13(4,)\chi_{13} (4, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 13, (0: ), 0.859+0.511i)(1,\ 13,\ (0:\ ),\ 0.859 + 0.511i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.5439933202+0.1495074800i0.5439933202 + 0.1495074800i
L(12)L(\frac12) \approx 0.5439933202+0.1495074800i0.5439933202 + 0.1495074800i
L(1)L(1) \approx 0.8231273762+0.1859280777i0.8231273762 + 0.1859280777i
L(1)L(1) \approx 0.8231273762+0.1859280777i0.8231273762 + 0.1859280777i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad13 1 1
good2 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
5 1T 1 - T
7 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
17 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
19 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
23 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
29 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
31 1T 1 - T
37 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
41 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
43 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
47 1T 1 - T
53 1+T 1 + T
59 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
61 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
67 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
71 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
73 1T 1 - T
79 1+T 1 + T
83 1T 1 - T
89 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
97 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−44.042538338572824817320349453367, −42.4381505634629781486590400547, −40.54253789223337291520536445179, −39.67695817814093113780009353001, −38.34988310323908670565844503340, −37.54058535341490773551060114884, −35.24321469346156156202293284029, −33.70972897257318246712380632676, −32.04372695737331039705868433686, −31.15408393811924641483439120288, −29.276524014363776500781207107028, −27.784285780986644537584886981993, −27.104781173085398003662802468647, −24.17186681129930729002901535625, −22.68077176274483011651309367952, −21.57530316919110316790947925575, −20.09754801517745622026998311215, −18.382979911139294498264833654699, −15.96780220625727028475050807624, −14.56208942822243278659626643472, −11.99504392637782327338231052927, −11.12452605047154763029618405683, −9.04699334806704048478002090503, −5.42273850072170533908335115787, −3.66097464838443443947257510998, 4.454853911564068083703595038243, 6.80983106916894389432472660976, 7.99512842484801144906191271642, 11.53031305636338257646752187580, 13.03559042617913789432092819792, 14.77690032158991012554008142959, 16.64931851786381053385281175531, 17.93154649566368497638434173782, 19.95742714793315041173639702469, 22.46606086380455732657855395522, 23.56721734639499833998388243743, 24.47347955260311207992017331125, 26.34106658141893115588573031050, 27.94708686509299663405920599574, 30.26856008929587796877185064518, 30.92121120986112335976162807961, 32.88015347184511555416500223370, 34.278212085285161671628017139637, 35.37863743350305940833066011129, 36.39790260310456600503519815589, 39.158077373401764597590580905046, 40.12967928685814949328369040309, 41.47132855075354572950992974996, 42.67809326968208595249856242977, 43.70125476149517110116121945808

Graph of the ZZ-function along the critical line