Properties

Label 2-462-7.4-c1-0-2
Degree $2$
Conductor $462$
Sign $0.905 - 0.424i$
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 + (−1.37 − 2.37i)5-s + 0.999·6-s + (−1.37 + 2.26i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.37 + 2.37i)10-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)12-s + 5.49·13-s + (2.64 + 0.0585i)14-s + 2.74·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.614 − 1.06i)5-s + 0.408·6-s + (−0.519 + 0.854i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.434 + 0.752i)10-s + (−0.150 + 0.261i)11-s + (−0.144 − 0.249i)12-s + 1.52·13-s + (0.706 + 0.0156i)14-s + 0.709·15-s + (−0.125 − 0.216i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.905 - 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812488 + 0.180869i\)
\(L(\frac12)\) \(\approx\) \(0.812488 + 0.180869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 - 2.26i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.37 + 2.37i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.49T + 13T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.01 - 6.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.645 - 1.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.54T + 29T^{2} \)
31 \( 1 + (2.54 - 4.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.01 - 3.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.54T + 41T^{2} \)
43 \( 1 + 4.03T + 43T^{2} \)
47 \( 1 + (-4.64 - 8.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.74 - 4.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.76 + 8.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.829 - 1.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.77 - 3.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.54T + 71T^{2} \)
73 \( 1 + (-4.29 + 7.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.11 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (-3.25 - 5.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.0872T + 97T^{2} \)
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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

−11.15520953042216874453511955779, −10.19473009543171739105848848372, −9.300485304616399899543842208427, −8.613037048755244844015138392851, −7.903587652300124539485191111388, −6.24677609641353778251111415590, −5.29357517138920473619112426587, −4.14329712253266377842133933876, −3.21336850420293364127955590820, −1.28662519061894609373349649413, 0.71633729671301642371327671941, 2.95982745103062797297962047565, 4.10957094732783205129893072402, 5.64111985714492784038539891908, 6.73820776912544181274593874587, 7.06981328901468869820216409737, 8.021239511651636772415100266299, 9.054782097399353921593833571345, 10.21441544788919978369206596489, 11.09658571099216504256787618546

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