Properties

Label 1-69-69.65-r0-0-0
Degree $1$
Conductor $69$
Sign $0.551 - 0.833i$
Analytic cond. $0.320434$
Root an. cond. $0.320434$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.551 - 0.833i$
Analytic conductor: \(0.320434\)
Root analytic conductor: \(0.320434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (0:\ ),\ 0.551 - 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157781089 - 0.6222501876i\)
\(L(\frac12)\) \(\approx\) \(1.157781089 - 0.6222501876i\)
\(L(1)\) \(\approx\) \(1.296875136 - 0.5088334495i\)
\(L(1)\) \(\approx\) \(1.296875136 - 0.5088334495i\)

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

−31.90196696819895031004113725494, −31.33100335424952048233845816779, −30.10396598511044029807649260263, −28.79162626756989133616751105070, −27.57795948667380412248841899142, −26.36181742206216270388533207061, −25.09040982992695706456377771010, −24.392484013159481372640512186241, −23.49768109488977449719674438798, −22.05484257531482599798541893765, −21.10821372041451841569765678632, −20.202693838548457598344606626913, −18.042581850945855984628380721176, −17.32316857388837539613312938315, −16.08384450774100229225129945640, −14.96166860094241094860490635207, −13.80707757191050284850687200470, −12.670574100628375912779505096359, −11.62675203339405445223072753834, −9.508643162531325527786034256048, −8.2664969767944037731636362412, −7.04430307491684680065018181079, −5.16925208075742031266845119264, −4.731174009906145718874668847521, −2.369749188700027756309953957192, 1.92587533093533017326671747685, 3.32936498685472449528913005891, 4.97978559407569485605474266115, 6.26338745614140619578281007109, 8.029664593333413064522187819895, 10.03931537945080067278740281861, 10.77316685640013839343320452079, 11.96245140035995952220777436812, 13.46768021695720713980922537487, 14.379917382032631124076726206555, 15.2839287829136114108530413021, 17.30556333358272707364139058477, 18.4568166120989675148478549980, 19.43551964895529780288156805192, 20.88827762451715203163225146269, 21.59995462904383335884074173337, 22.67155636824673234016363963094, 23.80760151540916386331254168808, 24.79907913051291896184364385424, 26.527115868335944188100033570683, 27.32499803656878843653781605346, 28.78621637605325088501860360882, 29.69925606885180454421527035572, 30.48685527545157827791791679415, 31.46725487108747817436356941779

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