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Properties

Label 2-462-7.2-c1-0-4
Degree $2$
Conductor $462$
Sign $-0.273 - 0.961i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.806 − 1.39i)5-s − 0.999·6-s + (−0.806 + 2.51i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.806 + 1.39i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−1.77 − 1.95i)14-s + 1.61·15-s + (−0.5 + 0.866i)16-s + (3.67 + 6.35i)17-s + (−0.499 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.360 − 0.624i)5-s − 0.408·6-s + (−0.304 + 0.952i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.255 + 0.441i)10-s + (0.150 + 0.261i)11-s + (0.144 − 0.249i)12-s + (−0.475 − 0.523i)14-s + 0.416·15-s + (−0.125 + 0.216i)16-s + (0.890 + 1.54i)17-s + (−0.117 − 0.204i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.273 - 0.961i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.273 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750768 + 0.994147i\)
\(L(\frac12)\) \(\approx\) \(0.750768 + 0.994147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.806 - 2.51i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.806 + 1.39i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-3.67 - 6.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.585 - 1.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.77 - 3.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + (3.17 + 5.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.41 - 2.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.94T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (3.39 - 5.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4 + 6.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.19 + 3.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.97 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.47 + 2.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 + (-2.55 - 4.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.80 - 4.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.05T + 83T^{2} \)
89 \( 1 + (-0.386 + 0.669i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−11.16621498681326601354058810937, −9.906931078164251771196487250542, −9.611386231703163205880920218647, −8.481963637479144886512688406776, −8.084568724400794936776106037696, −6.52626991346947614902147109602, −5.70074183232271459602453796950, −4.83088927317290942119420186579, −3.46121201806650984943637691595, −1.76492509313513827125164267848, 0.900292199053185523838336862067, 2.58389578842827060809559373832, 3.44190066245201312876857889554, 4.85980901983153694818112454399, 6.46960265093436413061249155922, 7.10948005661478646724914787101, 8.087678554729085407101998941639, 9.101242528311162601091515502432, 10.11399205752708951932981052304, 10.55819713621662013573374569227

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