Properties

Degree 1
Conductor 5
Sign $0.850 + 0.525i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $0.850 + 0.525i$
motivic weight  =  \(0\)
character  :  $\chi_{5} (2, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 5,\ (1:\ ),\ 0.850 + 0.525i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7637478801 + 0.2169647675i$
$L(\frac12,\chi)$  $\approx$  $0.7637478801 + 0.2169647675i$
$L(\chi,1)$  $\approx$  0.8648062659 + 0.2041530661i
$L(1,\chi)$  $\approx$  0.8648062659 + 0.2041530661i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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