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Properties

Label 1-469-469.207-r0-0-0
Degree $1$
Conductor $469$
Sign $-0.987 + 0.156i$
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  + (0.928 − 0.371i)2-s + (−0.327 + 0.945i)3-s + (0.723 − 0.690i)4-s + (−0.888 + 0.458i)5-s + (0.0475 + 0.998i)6-s + (0.415 − 0.909i)8-s + (−0.786 − 0.618i)9-s + (−0.654 + 0.755i)10-s + (−0.888 + 0.458i)11-s + (0.415 + 0.909i)12-s + (−0.995 − 0.0950i)13-s + (−0.142 − 0.989i)15-s + (0.0475 − 0.998i)16-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)18-s + (−0.786 + 0.618i)19-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (−0.327 + 0.945i)3-s + (0.723 − 0.690i)4-s + (−0.888 + 0.458i)5-s + (0.0475 + 0.998i)6-s + (0.415 − 0.909i)8-s + (−0.786 − 0.618i)9-s + (−0.654 + 0.755i)10-s + (−0.888 + 0.458i)11-s + (0.415 + 0.909i)12-s + (−0.995 − 0.0950i)13-s + (−0.142 − 0.989i)15-s + (0.0475 − 0.998i)16-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)18-s + (−0.786 + 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ -0.987 + 0.156i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02768044679 + 0.3516271270i\)
\(L(\frac12)\) \(\approx\) \(0.02768044679 + 0.3516271270i\)
\(L(1)\) \(\approx\) \(0.9209985006 + 0.1630175079i\)
\(L(1)\) \(\approx\) \(0.9209985006 + 0.1630175079i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.928 - 0.371i)T \)
3 \( 1 + (-0.327 + 0.945i)T \)
5 \( 1 + (-0.888 + 0.458i)T \)
11 \( 1 + (-0.888 + 0.458i)T \)
13 \( 1 + (-0.995 - 0.0950i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (-0.327 + 0.945i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.580 + 0.814i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.723 + 0.690i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.235 + 0.971i)T \)
59 \( 1 + (-0.995 + 0.0950i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (0.235 + 0.971i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (-0.888 + 0.458i)T \)
89 \( 1 + (-0.327 - 0.945i)T \)
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

−23.69112214492310266013106794728, −22.71201707838407250218137760574, −22.13570758539686931676908394182, −20.93539957688704069784613859205, −20.02662323585667614861364402918, −19.36062592141221235483111904053, −18.31171238154003598004201493225, −17.19205700240062600942886099466, −16.541650059134675369683041674563, −15.61811885029395124434664012871, −14.792980831733111824292531848716, −13.67440266974669277047767065801, −12.92837203626219688536457087842, −12.321531670401363936574488198673, −11.45205073592089942009045295211, −10.72524216954663235243020893299, −8.72521705603298165636638882457, −7.9730888352456669001525528576, −7.16351765300351808355983755182, −6.34595457460099989303223486857, −5.114431875012910501132876690725, −4.517505946384644408313596804532, −3.02153245537109079925464889807, −2.09589741744596892913675824306, −0.132625659579325955851247907731, 2.22520387101884903357270699473, 3.24323069108937950073472011302, 4.21605991319965339507042914186, 4.85473638865404909216170919401, 5.941896514917533034155346801729, 7.02230704726835799484031197482, 8.10085003598928796282070203496, 9.63681892757184289043014769521, 10.4385040665950886535767038150, 11.1046868636408722925725343284, 11.980435020506668409796790245685, 12.725695806580935297584952718704, 13.99758276524070766912191350112, 15.052910845274739496288061117339, 15.31202984638297186428180751305, 16.14074047662114201668387132614, 17.23515751582306818862304935423, 18.403130096461409345497808673508, 19.65850110694510232042176542676, 19.96906537409369725369679257357, 21.20192953257907624526586356048, 21.640679542552322175880450414767, 22.72651316198528285485057649135, 23.07313296981216768851186626564, 23.882458711891579851951913374514

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