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Properties

Label 1-65-65.29-r0-0-0
Degree $1$
Conductor $65$
Sign $-0.252 - 0.967i$
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 + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 12-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 18-s + (−0.5 − 0.866i)19-s + 21-s + (0.5 + 0.866i)22-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 12-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 18-s + (−0.5 − 0.866i)19-s + 21-s + (0.5 + 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7627970948 - 0.9875251257i\)
\(L(\frac12)\) \(\approx\) \(0.7627970948 - 0.9875251257i\)
\(L(1)\) \(\approx\) \(1.050035277 - 0.8363578568i\)
\(L(1)\) \(\approx\) \(1.050035277 - 0.8363578568i\)

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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−32.42010862813176179833337357864, −31.76920269969588802185622539971, −30.688072831917224724826193328946, −29.46336158149335837153007320620, −27.47150039501942720659128883644, −26.9043000576392578238379075452, −25.902425012526809853275431578332, −24.82607172852296434990571328202, −23.6013123373186392157547518483, −22.58701917558073632783094143150, −21.24611505336060201207803671635, −20.67332901901160763387043870791, −18.917743847966046999149626223413, −17.234766539022306065417540962757, −16.38090023426734555479224792778, −15.296014174032121372593967192560, −14.13959855167208589275345054345, −13.43017943855017145124775016787, −11.46305480388292347207756654635, −9.99217075936485430033068526732, −8.48323352804523311902607402178, −7.520789659615624980152397915171, −5.63247449588784096696620852951, −4.39887009262810043290435386575, −3.175537039960978274481755874814, 1.75567166832754337999448077034, 2.89148947264046073425478603078, 4.80816607467175370693244735981, 6.33989762789827850612300784517, 8.153873098932425659004863878980, 9.38236478841637930060054492697, 11.023744021614194309581559136201, 12.36094176547798086321051028618, 12.97129302610410332320714807, 14.48311524930716596512906510233, 15.183650418534814075129505652393, 17.6403640946611194017085445020, 18.52594060642607074333210542743, 19.50030876721520043030308917113, 20.62620334763276376747955635713, 21.57907239917921341815126493217, 23.02267440980809811938924204746, 23.96399310436246525151189479693, 24.99647798758030871491865685915, 26.28181606368490972456885342893, 27.96929474635040755160115442345, 28.63773565111887949869213734405, 29.98324516341783548448702668817, 30.73781283436154842540393239148, 31.487602025094913433214974443007

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