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Properties

Label 1-69-69.32-r1-0-0
Degree $1$
Conductor $69$
Sign $0.0230 - 0.999i$
Analytic cond. $7.41507$
Root an. cond. $7.41507$
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  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (0.959 − 0.281i)5-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (0.959 + 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (0.142 − 0.989i)20-s + 22-s + (0.841 − 0.540i)25-s + (0.142 − 0.989i)26-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (0.959 − 0.281i)5-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (0.959 + 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (0.142 − 0.989i)20-s + 22-s + (0.841 − 0.540i)25-s + (0.142 − 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.0230 - 0.999i$
Analytic conductor: \(7.41507\)
Root analytic conductor: \(7.41507\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (1:\ ),\ 0.0230 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5281609710 - 0.5404887979i\)
\(L(\frac12)\) \(\approx\) \(0.5281609710 - 0.5404887979i\)
\(L(1)\) \(\approx\) \(0.6860242818 - 0.09579848749i\)
\(L(1)\) \(\approx\) \(0.6860242818 - 0.09579848749i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 - T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−31.65993748406984448975488168705, −30.54943802432003717922036087561, −29.28235487549766760668450538091, −28.81409268808544708563359436818, −27.657558527827079062604552239421, −26.27834022147104042838782681657, −25.57849546402114274753893380822, −24.65123279923999774030954314473, −22.60271916891602403414588395192, −21.73642657812398332899472551977, −20.70324466716484378479752161212, −19.45991164483348561712894587811, −18.29970004447684786822303145738, −17.59541026800249944146247769514, −16.25351191435190326148844569503, −14.95922189510995800674575129804, −13.08725374761155825107817505447, −12.326686269054747713845290642156, −10.52616938068108845469061047743, −9.84768287018203779607747834305, −8.570558574320425350737108753176, −7.03821610171147938511036762766, −5.54125920274232410552090092139, −3.119929083412549940294630492375, −1.94166751396486846214629632492, 0.47580111854292612979898858379, 2.41602522775930174864492171405, 4.98957174502183185135608928808, 6.35972046139459242203019833544, 7.50181903364466193965463509791, 9.16808337914729649436642873908, 9.91582469184116634122560162260, 11.20664101782339476838768415615, 13.247459941380982361888064144850, 14.12568858996884696051556848663, 15.79619419069244483594717043892, 16.70478321884449542268603135222, 17.647774641841628145667312856861, 18.81360500694868466098914698390, 19.98318994483721091480918114928, 21.10061060911260827890682989456, 22.60643960702298309384211347768, 24.00686263844853232262834455710, 24.75769246372188253831238497624, 26.27578240112032602173285903566, 26.38987275859216500912488450462, 28.1020280871733359573396962194, 29.16494903267967708344182007544, 29.578531117385949296912896922608, 31.65604592086946234506889641669

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