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Properties

Label 2-469-469.160-c0-0-0
Degree $2$
Conductor $469$
Sign $0.318 + 0.948i$
Analytic cond. $0.234061$
Root an. cond. $0.483799$
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  + (−0.165 − 0.231i)2-s + (0.300 − 0.868i)4-s + (−0.995 + 0.0950i)7-s + (−0.524 + 0.153i)8-s + (0.415 − 0.909i)9-s + (1.07 − 0.431i)11-s + (0.186 + 0.215i)14-s + (−0.600 − 0.471i)16-s + (−0.279 + 0.0538i)18-s + (−0.277 − 0.178i)22-s + (0.0224 + 0.470i)23-s + (−0.959 − 0.281i)25-s + (−0.216 + 0.893i)28-s + (0.995 + 1.72i)29-s + (−0.0363 + 0.762i)32-s + ⋯
L(s)  = 1  + (−0.165 − 0.231i)2-s + (0.300 − 0.868i)4-s + (−0.995 + 0.0950i)7-s + (−0.524 + 0.153i)8-s + (0.415 − 0.909i)9-s + (1.07 − 0.431i)11-s + (0.186 + 0.215i)14-s + (−0.600 − 0.471i)16-s + (−0.279 + 0.0538i)18-s + (−0.277 − 0.178i)22-s + (0.0224 + 0.470i)23-s + (−0.959 − 0.281i)25-s + (−0.216 + 0.893i)28-s + (0.995 + 1.72i)29-s + (−0.0363 + 0.762i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.318 + 0.948i$
Analytic conductor: \(0.234061\)
Root analytic conductor: \(0.483799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 469,\ (\ :0),\ 0.318 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8011710626\)
\(L(\frac12)\) \(\approx\) \(0.8011710626\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
good2 \( 1 + (0.165 + 0.231i)T + (-0.327 + 0.945i)T^{2} \)
3 \( 1 + (-0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (-1.07 + 0.431i)T + (0.723 - 0.690i)T^{2} \)
13 \( 1 + (-0.0475 + 0.998i)T^{2} \)
17 \( 1 + (0.786 - 0.618i)T^{2} \)
19 \( 1 + (-0.981 - 0.189i)T^{2} \)
23 \( 1 + (-0.0224 - 0.470i)T + (-0.995 + 0.0950i)T^{2} \)
29 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.0475 - 0.998i)T^{2} \)
37 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.928 - 0.371i)T^{2} \)
43 \( 1 + (1.28 - 1.48i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-0.580 - 0.814i)T^{2} \)
53 \( 1 + (-1.16 - 1.34i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.723 - 0.690i)T^{2} \)
71 \( 1 + (0.607 - 1.75i)T + (-0.786 - 0.618i)T^{2} \)
73 \( 1 + (-0.723 - 0.690i)T^{2} \)
79 \( 1 + (-0.396 - 1.63i)T + (-0.888 + 0.458i)T^{2} \)
83 \( 1 + (-0.235 - 0.971i)T^{2} \)
89 \( 1 + (-0.415 - 0.909i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.03671995972431903179208826982, −10.00725502519939393006162507798, −9.455189929686518179920755591530, −8.747744969781626400988847996454, −7.07898685230389057204921779609, −6.41254972529302786941297816901, −5.67593915870198441826855864431, −4.09680779754112046917518735246, −2.98249364852374220239573148599, −1.25280790371751161020699070125, 2.24635860648660622821238047035, 3.56358491623507833631539345254, 4.51533352097538473875503055017, 6.16497185040842784106345082094, 6.85887642343136479662004470868, 7.73647249532345995783203695869, 8.629157859571475389578094294081, 9.661447226220847173167730105961, 10.34648706392526356073003838775, 11.73723664936513328207149886363

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