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Properties

Label 1-469-469.115-r0-0-0
Degree $1$
Conductor $469$
Sign $0.178 - 0.983i$
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  + (0.995 − 0.0950i)2-s + (−0.888 − 0.458i)3-s + (0.981 − 0.189i)4-s + (−0.786 − 0.618i)5-s + (−0.928 − 0.371i)6-s + (0.959 − 0.281i)8-s + (0.580 + 0.814i)9-s + (−0.841 − 0.540i)10-s + (0.786 + 0.618i)11-s + (−0.959 − 0.281i)12-s + (0.723 − 0.690i)13-s + (0.415 + 0.909i)15-s + (0.928 − 0.371i)16-s + (0.654 − 0.755i)17-s + (0.654 + 0.755i)18-s + (−0.580 + 0.814i)19-s + ⋯
L(s)  = 1  + (0.995 − 0.0950i)2-s + (−0.888 − 0.458i)3-s + (0.981 − 0.189i)4-s + (−0.786 − 0.618i)5-s + (−0.928 − 0.371i)6-s + (0.959 − 0.281i)8-s + (0.580 + 0.814i)9-s + (−0.841 − 0.540i)10-s + (0.786 + 0.618i)11-s + (−0.959 − 0.281i)12-s + (0.723 − 0.690i)13-s + (0.415 + 0.909i)15-s + (0.928 − 0.371i)16-s + (0.654 − 0.755i)17-s + (0.654 + 0.755i)18-s + (−0.580 + 0.814i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.178 - 0.983i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.178 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368182318 - 1.142617906i\)
\(L(\frac12)\) \(\approx\) \(1.368182318 - 1.142617906i\)
\(L(1)\) \(\approx\) \(1.327866036 - 0.5235216294i\)
\(L(1)\) \(\approx\) \(1.327866036 - 0.5235216294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.995 - 0.0950i)T \)
3 \( 1 + (-0.888 - 0.458i)T \)
5 \( 1 + (-0.786 - 0.618i)T \)
11 \( 1 + (0.786 + 0.618i)T \)
13 \( 1 + (0.723 - 0.690i)T \)
17 \( 1 + (0.654 - 0.755i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.235 - 0.971i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.981 + 0.189i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.841 + 0.540i)T \)
53 \( 1 + (0.327 - 0.945i)T \)
59 \( 1 + (-0.723 - 0.690i)T \)
61 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.327 + 0.945i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.786 + 0.618i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
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

−23.69790082224658534146490722762, −23.27421066267984308558036886000, −22.32643454916266314871029122747, −21.7329979154223396927003571535, −21.09111261783379502629288584889, −19.81344766733937226276675149713, −19.12544341656307688065766178121, −17.947642561220304770403751087817, −16.78622479667257984842740138423, −16.20889150376106229558921205471, −15.394954757309767858867400175724, −14.621743431250738531407852212133, −13.74021827215822087403284066510, −12.45011089143169210059581837642, −11.79608811822305754883103877677, −11.04614015639270048609427739567, −10.47730979259430516850371139538, −8.93026774596711833922059431072, −7.61725211314483961073670510300, −6.50293930010562055255166860173, −6.11380120151570093804272495649, −4.776145267782569374004155665231, −3.89241431897188210907191227987, −3.27890955872546497790166870377, −1.457843666512035913942751272183, 0.89476631348787155237384779875, 2.07621328191550664624595459479, 3.75603423099271191745786165868, 4.41898299817283109068213026065, 5.523082924786973640583293136083, 6.24509956548118153208876427101, 7.39112899262121685789935567820, 8.07461941922723479364368431556, 9.76625623531738409841880250747, 10.87379931823745880248073606792, 11.688857187460672353204741010422, 12.33234226805847739785187272803, 12.90822692065710228226107324006, 13.9797139486193956448768521801, 15.04072824353278115771973169748, 15.98529838731271637303667994212, 16.55274611076145787433337501066, 17.47999909273435027251055423257, 18.716602160466814788110476358767, 19.50183531854599020410307116065, 20.47707917985167764142283867718, 21.0747979556898150204402504251, 22.465174526641204154237752989261, 22.83358261830352817207111134757, 23.41406535070233704796572618962

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