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Properties

Label 1-13-13.4-r0-0-0
Degree $1$
Conductor $13$
Sign $0.859 + 0.511i$
Analytic cond. $0.0603717$
Root an. cond. $0.0603717$
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  + (0.5 + 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

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

Invariants

Degree: \(1\)
Conductor: \(13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.0603717\)
Root analytic conductor: \(0.0603717\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 13,\ (0:\ ),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5439933202 + 0.1495074800i\)
\(L(\frac12)\) \(\approx\) \(0.5439933202 + 0.1495074800i\)
\(L(1)\) \(\approx\) \(0.8231273762 + 0.1859280777i\)
\(L(1)\) \(\approx\) \(0.8231273762 + 0.1859280777i\)

Euler product

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

−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 $Z$-function along the critical line