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Properties

Label 1-65-65.33-r0-0-0
Degree $1$
Conductor $65$
Sign $0.00418 - 0.999i$
Analytic cond. $0.301858$
Root an. cond. $0.301858$
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)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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.00418 - 0.999i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ 0.00418 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6373996914 - 0.6400731312i\)
\(L(\frac12)\) \(\approx\) \(0.6373996914 - 0.6400731312i\)
\(L(1)\) \(\approx\) \(0.8381921562 - 0.5311579799i\)
\(L(1)\) \(\approx\) \(0.8381921562 - 0.5311579799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−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