Properties

Label 1-5-5.2-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: $\chi_{5} (2, \cdot )$
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

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

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