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Properties

Label 1-469-469.121-r0-0-0
Degree $1$
Conductor $469$
Sign $0.991 - 0.132i$
Analytic cond. $2.17802$
Root an. cond. $2.17802$
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  + (−0.959 + 0.281i)2-s + (0.928 + 0.371i)3-s + (0.841 − 0.540i)4-s + (−0.995 + 0.0950i)5-s + (−0.995 − 0.0950i)6-s + (−0.654 + 0.755i)8-s + (0.723 + 0.690i)9-s + (0.928 − 0.371i)10-s + (0.415 − 0.909i)11-s + (0.981 − 0.189i)12-s + (0.981 − 0.189i)13-s + (−0.959 − 0.281i)15-s + (0.415 − 0.909i)16-s + (0.0475 − 0.998i)17-s + (−0.888 − 0.458i)18-s + (−0.959 − 0.281i)19-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (0.928 + 0.371i)3-s + (0.841 − 0.540i)4-s + (−0.995 + 0.0950i)5-s + (−0.995 − 0.0950i)6-s + (−0.654 + 0.755i)8-s + (0.723 + 0.690i)9-s + (0.928 − 0.371i)10-s + (0.415 − 0.909i)11-s + (0.981 − 0.189i)12-s + (0.981 − 0.189i)13-s + (−0.959 − 0.281i)15-s + (0.415 − 0.909i)16-s + (0.0475 − 0.998i)17-s + (−0.888 − 0.458i)18-s + (−0.959 − 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.991 - 0.132i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.991 - 0.132i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036181333 - 0.06886832089i\)
\(L(\frac12)\) \(\approx\) \(1.036181333 - 0.06886832089i\)
\(L(1)\) \(\approx\) \(0.8727355017 + 0.07306111305i\)
\(L(1)\) \(\approx\) \(0.8727355017 + 0.07306111305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (0.928 + 0.371i)T \)
5 \( 1 + (-0.995 + 0.0950i)T \)
11 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (0.981 - 0.189i)T \)
17 \( 1 + (0.0475 - 0.998i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (0.0475 - 0.998i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.928 + 0.371i)T \)
53 \( 1 + (0.0475 + 0.998i)T \)
59 \( 1 + (-0.327 - 0.945i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.888 - 0.458i)T \)
73 \( 1 + (0.580 - 0.814i)T \)
79 \( 1 + (0.981 - 0.189i)T \)
83 \( 1 + (0.580 + 0.814i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
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

−23.905810460365827003490089836787, −23.40459205891052204148397531314, −21.90448197714472461583414152282, −20.91270233294977746047872408820, −20.19028776769834518567200711449, −19.6111334616658085526358495472, −18.885178569059726206350908815494, −18.1765479078548407058457056475, −17.12237223579743862328045395437, −16.13663467601311445441542650308, −15.2024354898003399207537434886, −14.717240818295438978101541217516, −13.103458490961109180427138479587, −12.5062674934836272655143678037, −11.534686849252618644404903830349, −10.60259707319726912616503088643, −9.450014683220026817162131925788, −8.702153360669630454707072668261, −7.93069661071135187115845807954, −7.220133325061448433298260136404, −6.242882776296652553434821889251, −4.05967585343269515433791927055, −3.59487932227327021431185586915, −2.16336593464654933889031269866, −1.25102620446639585925440441528, 0.818109778554338473513651773359, 2.40976622788304301126934825102, 3.39939506638293230546788631770, 4.45004917768330773140569514198, 6.00109523362478557910552139810, 7.10832536725734354274237885375, 7.98940836874875073602951739801, 8.68869411572476534226286078041, 9.30567166581048287895478052451, 10.711818092912990287680667319913, 11.07691956183330453246420927692, 12.33082800915186069091060594188, 13.68943117158130360649666063229, 14.58460416177333784762194942567, 15.37682555139336268216860784786, 16.12259677556449708681657691557, 16.64161946202887393010973724567, 18.107515944220754120341936551711, 18.94366153842824418091680741448, 19.352205623170174028521301307967, 20.37102882435421585459947029710, 20.81954547457298037830904895240, 22.04762068753522747959843601673, 23.22528180968168490928339247152, 24.104998360263152549133502624157

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