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Properties

Label 2-469-469.419-c0-0-0
Degree $2$
Conductor $469$
Sign $0.929 - 0.369i$
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  + (0.283 − 0.0270i)2-s + (−0.902 + 0.173i)4-s + (0.580 + 0.814i)7-s + (−0.524 + 0.153i)8-s + (0.415 − 0.909i)9-s + (1.56 + 1.23i)11-s + (0.186 + 0.215i)14-s + (0.708 − 0.283i)16-s + (0.0930 − 0.268i)18-s + (0.476 + 0.306i)22-s + (−1.28 − 0.663i)23-s + (−0.959 − 0.281i)25-s + (−0.665 − 0.634i)28-s + (−0.580 + 1.00i)29-s + (0.678 − 0.349i)32-s + ⋯
L(s)  = 1  + (0.283 − 0.0270i)2-s + (−0.902 + 0.173i)4-s + (0.580 + 0.814i)7-s + (−0.524 + 0.153i)8-s + (0.415 − 0.909i)9-s + (1.56 + 1.23i)11-s + (0.186 + 0.215i)14-s + (0.708 − 0.283i)16-s + (0.0930 − 0.268i)18-s + (0.476 + 0.306i)22-s + (−1.28 − 0.663i)23-s + (−0.959 − 0.281i)25-s + (−0.665 − 0.634i)28-s + (−0.580 + 1.00i)29-s + (0.678 − 0.349i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.929 - 0.369i$
Analytic conductor: \(0.234061\)
Root analytic conductor: \(0.483799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 469,\ (\ :0),\ 0.929 - 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9064509808\)
\(L(\frac12)\) \(\approx\) \(0.9064509808\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.580 - 0.814i)T \)
67 \( 1 + (-0.928 + 0.371i)T \)
good2 \( 1 + (-0.283 + 0.0270i)T + (0.981 - 0.189i)T^{2} \)
3 \( 1 + (-0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (-1.56 - 1.23i)T + (0.235 + 0.971i)T^{2} \)
13 \( 1 + (0.888 - 0.458i)T^{2} \)
17 \( 1 + (-0.928 - 0.371i)T^{2} \)
19 \( 1 + (0.327 + 0.945i)T^{2} \)
23 \( 1 + (1.28 + 0.663i)T + (0.580 + 0.814i)T^{2} \)
29 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.888 + 0.458i)T^{2} \)
37 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.786 - 0.618i)T^{2} \)
43 \( 1 + (-0.428 + 0.494i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.995 - 0.0950i)T^{2} \)
53 \( 1 + (0.0623 + 0.0719i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.235 + 0.971i)T^{2} \)
71 \( 1 + (1.54 - 0.297i)T + (0.928 - 0.371i)T^{2} \)
73 \( 1 + (-0.235 + 0.971i)T^{2} \)
79 \( 1 + (-1.21 + 1.16i)T + (0.0475 - 0.998i)T^{2} \)
83 \( 1 + (-0.723 + 0.690i)T^{2} \)
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

−11.70995101494106658238948009635, −10.20495450729284244175971091687, −9.254850900513371679029003929988, −8.943615138745397695959197513444, −7.71106929493771235793677964722, −6.56988081133560916404349946624, −5.56239836358190962707234413009, −4.36691549510720037531013439518, −3.75611783247860969015481572580, −1.86294339093555698853766874905, 1.46085677850117055015982965937, 3.68345339213178962244386008740, 4.27346450950258326583875829168, 5.43474621008412698546859852632, 6.39399192435951223332079666545, 7.73752291002473013278380956492, 8.405753836330330170487576558438, 9.481297285957696905393546847177, 10.21886305262581660527282976704, 11.29223442940078912713479379554

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