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Properties

Label 1-65-65.17-r1-0-0
Degree $1$
Conductor $65$
Sign $-0.886 + 0.462i$
Analytic cond. $6.98522$
Root an. cond. $6.98522$
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  + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + i·18-s + (−0.5 + 0.866i)19-s − 21-s + (−0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + i·18-s + (−0.5 + 0.866i)19-s − 21-s + (−0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.886 + 0.462i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ -0.886 + 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1430851451 + 0.5836864112i\)
\(L(\frac12)\) \(\approx\) \(0.1430851451 + 0.5836864112i\)
\(L(1)\) \(\approx\) \(0.4814200058 + 0.3061342289i\)
\(L(1)\) \(\approx\) \(0.4814200058 + 0.3061342289i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−30.83346395362043574301728160049, −30.11163406018842243975305450113, −29.23196594022520716411706224425, −28.1259131526935666944922425547, −27.31646233803405221857625899577, −26.23192281074283340456557020997, −24.62713083235856242023410106007, −23.937048182437109049322744006890, −22.27162202953397182280235637076, −21.35255106537014494638746629077, −19.92258485340093942036359284817, −18.894752628877394191633363645013, −17.67261537478834057788328739483, −17.11603932064746316911321942323, −15.84730056890954533772186244586, −13.80783144085297706752342409664, −12.44619183247702835962178820543, −11.2300565907423357244214557129, −10.64511521528514824812300062829, −8.78278991285225465897601098463, −7.5474282210882288787743504004, −6.26116501409082213448960202704, −4.25953145020906238816915988881, −1.96613788265165931069765654422, −0.47108991964819114539807616418, 1.67790011843073376429174685107, 4.58768321619286234742794114559, 5.83622760604458733970295178423, 7.161353555319259545335128787268, 8.745600686325298851839618232620, 9.94020435762973798068973452161, 11.13233649766080621910090023032, 12.145460890885197634796437256807, 14.49121274773444141092445510142, 15.41923308847014282593545380344, 16.5409135698675563845899324193, 17.69683292848140481816945608231, 18.24217291580373679131479492801, 19.943843749781430593457944598452, 21.11034514857137394562464263144, 22.47145671493658974960493067823, 23.658923955998136094944011930377, 24.65911888015117805103892793369, 25.84795024213292963770926586364, 27.27829916520530105767620995525, 27.68867440864127462561786127855, 28.72684696704912443684855371314, 29.80577300150334799278415768308, 31.42325728496939476378004387011, 32.95801610026909597357397195564

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