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Properties

Label 1-5-5.3-r1-0-0
Degree $1$
Conductor $5$
Sign $0.850 - 0.525i$
Analytic cond. $0.537324$
Root an. cond. $0.537324$
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  − i·2-s + i·3-s − 4-s + 6-s − i·7-s + i·8-s − 9-s + 11-s − i·12-s + i·13-s − 14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯
L(s)  = 1  − i·2-s + i·3-s − 4-s + 6-s − i·7-s + i·8-s − 9-s + 11-s − i·12-s + i·13-s − 14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.537324\)
Root analytic conductor: \(0.537324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5,\ (1:\ ),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7637478801 - 0.2169647675i\)
\(L(\frac12)\) \(\approx\) \(0.7637478801 - 0.2169647675i\)
\(L(1)\) \(\approx\) \(0.8648062659 - 0.2041530661i\)
\(L(1)\) \(\approx\) \(0.8648062659 - 0.2041530661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + iT \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 - iT \)
43 \( 1 + iT \)
47 \( 1 - T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + iT \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + iT \)
97 \( 1 - T \)
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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

−54.48544238876467911470039917979, −53.442232173354543204000399997353, −51.977053467572707629124054585044, −50.66421039080575037299754421231, −48.4778466442218740132847945792, −46.59016101776473881831044964340, −44.82617597081092363119666478355, −42.99208544275153854582429789755, −41.53645675792969665969385524901, −40.39611485175259003483748872182, −37.27195057455605008724509915557, −35.49089317885139349790895129816, −34.45722878527839758405756301004, −32.195159688892272026544832306389, −30.46364068840366112797004484191, −27.982756935693594324451001091893, −25.531186800433429601457551452466, −24.36527977540229805651909575745, −22.48758458302875002505567290925, −18.9985880416861449287245250119, −17.33780210685303969091451014241, −14.82502557032842825143025217404, −12.67494641701135578048229914508, −8.45722917442323072160535286274, −6.18357819545085391437751730970, 4.13290370521285159500191933156, 9.44293112972850911710026212431, 11.2828964415816001332254807924, 14.11546426656964617536066631834, 16.99590394259028444664793427768, 19.72905478631162605830813598374, 21.2830471577778699486763506284, 22.9655764347914803808601277723, 26.47278891481336832795849047572, 27.81247022179307495811051208168, 29.70278103479729044076360156773, 31.714038216978378088676011094547, 33.30774552092450944486851535240, 36.09652653725364935231283520536, 37.84761841624699497844836046640, 39.0916158639927091132955773909, 40.508934445122041603749872022008, 43.12543488567473830905726366337, 45.08582264962074326882242881556, 46.26641381346456449258634989372, 48.47765991878316956160896236195, 49.5201121831721842836352461905, 51.08998804633721267564942694432, 53.75667308835424588804456665093, 55.53255063917714331177635885562

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