Properties

Degree 1
Conductor $ 5 \cdot 13 $
Sign $0.00418 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 18-s + (0.866 + 0.5i)19-s i·21-s + (0.866 + 0.5i)22-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 18-s + (0.866 + 0.5i)19-s i·21-s + (0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.00418 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.00418 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(65\)    =    \(5 \cdot 13\)
\( \varepsilon \)  =  $0.00418 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{65} (33, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 65,\ (0:\ ),\ 0.00418 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6373996914 - 0.6400731312i$
$L(\frac12,\chi)$  $\approx$  $0.6373996914 - 0.6400731312i$
$L(\chi,1)$  $\approx$  0.8381921562 - 0.5311579799i
$L(1,\chi)$  $\approx$  0.8381921562 - 0.5311579799i

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

−32.4821528087572273399087035498, −31.60941273102563295523092974165, −30.70886243385155805759447884244, −28.7147038844939886860130173512, −27.82816610853747625793444489544, −26.62789939734861403623330659999, −26.04463431078442436372942456191, −24.72135701954408657085143399554, −24.1493526634194710185838851936, −22.38338442211988780352771733980, −21.26653125525130869055287624156, −19.90629026186729791346149688557, −18.76514118186054459180730446075, −17.79077597338447390191097105555, −16.11086187526962033338353268354, −15.43494657481268266301605929455, −14.370367219727900758554687674840, −13.23059973713010318547809986986, −11.032554623922665657355364092941, −9.67812845187044013637361906571, −8.57541967357438286617517068816, −7.74080900530159699733370712455, −5.84461303897186855158682586715, −4.50396431889060021289781895991, −2.3487445216765832424296261106, 1.51864073810637068580616856263, 2.99693145605024225727098452158, 4.49053086375616470630667705560, 7.252149529869749540946765988756, 8.085214292365618216626211601929, 9.47500783103501845316828402093, 10.64309828470510439895962754078, 12.11860970757299022991950276048, 13.32812480003250788993134274092, 14.188298585610585047050480399427, 15.97488748225526985104385799322, 17.71307635432712627514552538002, 18.30254410904907572840820495946, 19.79784522451496747876008351458, 20.35086132022016746811894954007, 21.35588687215254267188130993008, 22.97098821586307702240679169311, 24.20727231302155440748219495546, 25.62164634439721284605574757927, 26.501330073421047965355999505677, 27.35182186658488850076567840310, 28.902831939523193861397595807174, 29.69621315406219927382418617743, 30.864026664669325394512055812200, 31.30667419938621711152112333348

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