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Properties

Label 1-469-469.170-r0-0-0
Degree $1$
Conductor $469$
Sign $-0.966 - 0.256i$
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  + (0.723 − 0.690i)2-s + (−0.786 − 0.618i)3-s + (0.0475 − 0.998i)4-s + (0.580 − 0.814i)5-s + (−0.995 + 0.0950i)6-s + (−0.654 − 0.755i)8-s + (0.235 + 0.971i)9-s + (−0.142 − 0.989i)10-s + (0.580 − 0.814i)11-s + (−0.654 + 0.755i)12-s + (0.981 + 0.189i)13-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.841 − 0.540i)17-s + (0.841 + 0.540i)18-s + (0.235 − 0.971i)19-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)2-s + (−0.786 − 0.618i)3-s + (0.0475 − 0.998i)4-s + (0.580 − 0.814i)5-s + (−0.995 + 0.0950i)6-s + (−0.654 − 0.755i)8-s + (0.235 + 0.971i)9-s + (−0.142 − 0.989i)10-s + (0.580 − 0.814i)11-s + (−0.654 + 0.755i)12-s + (0.981 + 0.189i)13-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.841 − 0.540i)17-s + (0.841 + 0.540i)18-s + (0.235 − 0.971i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ -0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2229329164 - 1.708637619i\)
\(L(\frac12)\) \(\approx\) \(0.2229329164 - 1.708637619i\)
\(L(1)\) \(\approx\) \(0.8699985855 - 1.061623572i\)
\(L(1)\) \(\approx\) \(0.8699985855 - 1.061623572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.723 - 0.690i)T \)
3 \( 1 + (-0.786 - 0.618i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
11 \( 1 + (0.580 - 0.814i)T \)
13 \( 1 + (0.981 + 0.189i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.235 - 0.971i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.327 + 0.945i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.0475 + 0.998i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.142 + 0.989i)T \)
53 \( 1 + (-0.888 + 0.458i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.888 + 0.458i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.580 - 0.814i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
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

−24.05677454465916667012602066027, −23.04737840881232416385100050084, −22.728312963114763755949583430754, −21.95950169050711774955246487948, −21.09280196922843629459140232930, −20.54424083088001763402770808456, −18.83933495332547763493813194509, −17.84322301654689349079696155034, −17.33541290451374957596135607877, −16.42897489072978129476864676651, −15.55918268563050297767659113577, −14.76768881598589394185264070292, −14.10964998829865450616828671558, −12.96557304746701143838556507498, −12.02950380131441050819088646437, −11.21477817597517897964916176062, −10.15779959427991490674316053789, −9.34675350148136071237200131745, −7.892397970992617300909430985854, −6.88223795871705666485807869958, −5.901977237671699596496570659320, −5.58251100759935966274473877472, −4.00667672330085165009277567165, −3.573782609104706104050552557, −1.886021216507513376067957377725, 0.911138543673679740514619584229, 1.59432818368329163598257843495, 3.00628299288361483846088078025, 4.35668003339173384282130575415, 5.32345394123766860451017753237, 5.99326668613294592787996524095, 6.8546844662949722619852344146, 8.449210753615092715513273506451, 9.4136851202488616922137827680, 10.54816856031148706467019316097, 11.37224701207353648712984214886, 12.134232703014982660907717867192, 12.93016099282793506080667176395, 13.705813501518325324861150661694, 14.28086439462569173420388203199, 16.00031169135693624339056912679, 16.416446928958001988910448141736, 17.63679047097321340383590644962, 18.41925340715452041168966901400, 19.237482283431093944331472301043, 20.20516556359241031752472774548, 21.01855093639326495035166643706, 21.91337214024862670498155357055, 22.438146529961931017273057670046, 23.658299835617869767263586314206

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