Properties

Label 1-104-104.61-r0-0-0
Degree $1$
Conductor $104$
Sign $-0.252 + 0.967i$
Analytic cond. $0.482973$
Root an. cond. $0.482973$
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)3-s − 5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 21-s + (−0.5 − 0.866i)23-s + 25-s − 27-s + (0.5 + 0.866i)29-s + 31-s + (−0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s − 5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 21-s + (−0.5 − 0.866i)23-s + 25-s − 27-s + (0.5 + 0.866i)29-s + 31-s + (−0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(0.482973\)
Root analytic conductor: \(0.482973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (0:\ ),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5512847262 + 0.7136989931i\)
\(L(\frac12)\) \(\approx\) \(0.5512847262 + 0.7136989931i\)
\(L(1)\) \(\approx\) \(0.8477874006 + 0.4593680241i\)
\(L(1)\) \(\approx\) \(0.8477874006 + 0.4593680241i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
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

−29.705407270134012978408597291778, −28.64689773544053930107950935234, −27.0218468261073436319733812201, −26.59974403815828800238010900459, −25.198926543745115301679106040975, −24.27744859916826801611122737905, −23.34782178057327352095367175836, −22.552500994426656066777050993697, −20.77973713847720101505492447631, −19.73439057130018001173306386502, −19.260978558693066293949714776853, −18.05767607010868325139665588364, −16.690889637544991192160021246761, −15.66032809807942426928061042229, −14.208236028373690216634974205678, −13.47105955746550691186765603113, −12.148724318318191906642528053062, −11.258524197718244437665020164525, −9.562476740058880874337850835216, −8.19963661788668637866617978427, −7.349830454594368796872666702452, −6.21450542136112926092012397386, −4.07101029658962685950891096778, −3.02693120604489576413585062608, −0.90263017105314404117282313752, 2.53956168451872365177254757974, 3.846499361444573578708185295550, 4.928118873495006949239879847725, 6.70364381300278772246586253413, 8.2550868903497222363121493673, 9.120690287008776518754663785697, 10.35051960445323989587585577542, 11.655659553099844315299142300446, 12.69949587860829995777604493932, 14.3418731883481875157422314124, 15.402257400858155073859952808679, 15.82815403699149059485070732737, 17.22087586641948999326485528498, 18.797376279549169811990788189098, 19.75268461460064632571256063342, 20.46531564833304557727419754431, 21.99458759500820733873286531894, 22.44563217798018356591253226551, 23.8615833924764817968973644715, 25.113678981771638114434567172320, 26.04454866365307998081401692234, 26.97304705150029841955494998337, 28.05272845142627925502075510497, 28.53443817694127680633050797956, 30.565327158432871946955542944338

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