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Properties

Label 1-65-65.63-r0-0-0
Degree $1$
Conductor $65$
Sign $0.492 + 0.870i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5020545854 + 0.2927598172i\)
\(L(\frac12)\) \(\approx\) \(0.5020545854 + 0.2927598172i\)
\(L(1)\) \(\approx\) \(0.6340832894 + 0.2307340149i\)
\(L(1)\) \(\approx\) \(0.6340832894 + 0.2307340149i\)

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

−32.06617028117791826740606743717, −30.386729636191154363773044422330, −29.757673702087661982642999945265, −28.6723099987690610846548005992, −27.501280886099680065233146252069, −27.07501863886621488436176059859, −25.771705789124256129655871797975, −23.93492191863656244658031110687, −22.845682662250234001594437179272, −21.72511816235440408449357613484, −20.890456243853527267592350447537, −19.66608509253244086642437737544, −18.36435280199195212181960439227, −17.041038463251518143338573128492, −16.7174615979483688654626784770, −14.70945304558287566171479399686, −13.12482014197920587190465261330, −11.76600951502966923121853826346, −10.896280200407154727999219087863, −9.92026412748462010601658896461, −8.466959284092180134602457665168, −6.752772855612325635355313205321, −4.77009543915575024691855270037, −3.6019285325694071265558533603, −1.14987979973364678875722093021, 1.583946100603862447754057994016, 4.78489260370065901244633862672, 5.91455142060705543144792768697, 7.076074581609006490766260958824, 8.38605046119600403247532134051, 9.83065265437003055996474338190, 11.33469784377209204535561584744, 12.54180925336981192863539802201, 14.18108448882613899032465085646, 15.33701710110518867197952211813, 16.65342515228657912094698471376, 17.53443785885501991314287750396, 18.50376011299413323141589504045, 19.43915099430761982965989574776, 21.43954616121207862689612378490, 22.76622113698125987192703629067, 23.531229626411784388440293209076, 24.926083973597589965328376608017, 25.20983012472677324727812856982, 27.16293562617123959576338732256, 27.83280739277256314268110358235, 28.75224552574012018369914748573, 30.07548822569486216531922813175, 31.37729543204437756527712884928, 32.68807853609589652006260734895

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