Properties

Label 2-462-77.10-c1-0-3
Degree $2$
Conductor $462$
Sign $0.472 - 0.881i$
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  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−2.13 + 1.23i)5-s + 0.999·6-s + (0.941 − 2.47i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.23 − 2.13i)10-s + (−2.32 + 2.36i)11-s + (−0.866 + 0.499i)12-s + 1.32·13-s + (0.420 + 2.61i)14-s + 2.46·15-s + (−0.5 − 0.866i)16-s + (2.23 − 3.87i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.953 + 0.550i)5-s + 0.408·6-s + (0.356 − 0.934i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.389 − 0.674i)10-s + (−0.701 + 0.712i)11-s + (−0.249 + 0.144i)12-s + 0.366·13-s + (0.112 + 0.698i)14-s + 0.635·15-s + (−0.125 − 0.216i)16-s + (0.542 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.472 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584805 + 0.350167i\)
\(L(\frac12)\) \(\approx\) \(0.584805 + 0.350167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.941 + 2.47i)T \)
11 \( 1 + (2.32 - 2.36i)T \)
good5 \( 1 + (2.13 - 1.23i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.32T + 13T^{2} \)
17 \( 1 + (-2.23 + 3.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.21 - 3.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.14 - 7.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.44iT - 29T^{2} \)
31 \( 1 + (-2.34 - 1.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.46 - 4.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.18T + 41T^{2} \)
43 \( 1 - 9.63iT - 43T^{2} \)
47 \( 1 + (0.664 - 0.383i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.945 + 1.63i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.74 - 3.89i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.861 + 1.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (-5.58 + 9.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.53 - 1.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (10.9 - 6.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.37iT - 97T^{2} \)
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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

−11.27349661441202752976728176025, −10.32150422966945376178607503383, −9.590117162722729754161741706631, −8.061483519494731424158739394597, −7.47928723337314168060077058602, −7.03840506795827688864519937521, −5.65494980301138913527566783958, −4.56723092066253780815751358120, −3.19453430358606870725496619898, −1.19422142418489692507863836335, 0.66674791235821533085641364275, 2.65369244048533580370006682648, 4.00889143354406142143564802324, 5.12567622890408518215145881106, 6.13445076211404682744121139975, 7.52144038304987515562805358677, 8.474252617772503076241527815044, 8.844916212567382862124933964236, 10.13377301212994766240794034160, 11.06003251607283520453247138517

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