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Properties

Label 1-69-69.50-r1-0-0
Degree $1$
Conductor $69$
Sign $-0.999 + 0.0250i$
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  + (0.654 − 0.755i)2-s + (−0.142 − 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (−0.959 − 0.281i)10-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.841 + 0.540i)20-s + 22-s + (−0.654 + 0.755i)25-s + (−0.841 + 0.540i)26-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.142 − 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (−0.959 − 0.281i)10-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.841 + 0.540i)20-s + 22-s + (−0.654 + 0.755i)25-s + (−0.841 + 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.999 + 0.0250i$
Analytic conductor: \(7.41507\)
Root analytic conductor: \(7.41507\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (1:\ ),\ -0.999 + 0.0250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01594480590 - 1.272495347i\)
\(L(\frac12)\) \(\approx\) \(0.01594480590 - 1.272495347i\)
\(L(1)\) \(\approx\) \(0.7837627152 - 0.7685873589i\)
\(L(1)\) \(\approx\) \(0.7837627152 - 0.7685873589i\)

Euler product

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

−32.16344838284954587665783056038, −31.26469442806450195980079649119, −30.04339631030474722019608148535, −29.337733195682958425522369467391, −27.31724711616952529911063053136, −26.47445006493770391099542146705, −25.60513181663884056564485449024, −24.34674070358077011849598390880, −23.23900448479697628762224924040, −22.34995446552456728188311829149, −21.575388933814025159687439388786, −19.7439137645674778133185956490, −18.74075061197156891540275766270, −17.118828335972164244771336535112, −16.24239408969097248060333140893, −14.9374796744864671631532386940, −14.13253291384912185478999402853, −12.76391715899163987847729538844, −11.5794178488577670335041284259, −9.95843621077639631021960030837, −8.2291075202367752310088511715, −6.906072922648469822925374419008, −6.060143741768380347643686675204, −4.07462061953857083987613051575, −3.026984064622387764783315125939, 0.51307809705519744920004310773, 2.54014861416951217054938350095, 4.154275150075052354092237916, 5.306980499881969333617715184469, 6.97395205662791469023523599964, 9.08103237246441801027863302932, 9.92529773152285568214862800851, 11.75619872633928672757989594822, 12.44433488236117129511384663823, 13.5028138347521609123744817626, 15.0273612249198926735349714103, 16.045479103773865935839169112753, 17.569081949740993339935814076461, 19.308229163063601392944789402806, 19.827840674393859137580737095907, 20.9550254965658679386908291211, 22.28990000779017034752595968588, 23.01430154772086575279032811494, 24.33419472847554624393332950641, 25.20961794025842199812981198985, 27.04199218396655859066801616622, 28.108565978211339278654113702603, 28.85538862428073564103723275823, 29.95308631248031828432446791918, 31.14676286446958129704826692816

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