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Properties

Label 2-462-33.29-c1-0-9
Degree $2$
Conductor $462$
Sign $0.953 - 0.301i$
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  + (0.809 + 0.587i)2-s + (−0.979 − 1.42i)3-s + (0.309 + 0.951i)4-s + (0.571 + 0.786i)5-s + (0.0473 − 1.73i)6-s + (0.951 − 0.309i)7-s + (−0.309 + 0.951i)8-s + (−1.08 + 2.79i)9-s + 0.972i·10-s + (2.84 + 1.70i)11-s + (1.05 − 1.37i)12-s + (1.75 − 2.40i)13-s + (0.951 + 0.309i)14-s + (0.564 − 1.58i)15-s + (−0.809 + 0.587i)16-s + (1.52 − 1.10i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.565 − 0.824i)3-s + (0.154 + 0.475i)4-s + (0.255 + 0.351i)5-s + (0.0193 − 0.706i)6-s + (0.359 − 0.116i)7-s + (−0.109 + 0.336i)8-s + (−0.360 + 0.932i)9-s + 0.307i·10-s + (0.857 + 0.513i)11-s + (0.304 − 0.396i)12-s + (0.485 − 0.668i)13-s + (0.254 + 0.0825i)14-s + (0.145 − 0.409i)15-s + (−0.202 + 0.146i)16-s + (0.369 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.953 - 0.301i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.953 - 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79772 + 0.277040i\)
\(L(\frac12)\) \(\approx\) \(1.79772 + 0.277040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.979 + 1.42i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (-2.84 - 1.70i)T \)
good5 \( 1 + (-0.571 - 0.786i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.75 + 2.40i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.52 + 1.10i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.57 - 1.48i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.74iT - 23T^{2} \)
29 \( 1 + (1.68 + 5.18i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.90 + 1.38i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.449 - 1.38i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.867 + 2.67i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.83iT - 43T^{2} \)
47 \( 1 + (-3.10 - 1.00i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.54 - 6.26i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (5.00 - 1.62i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.75 + 6.55i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (8.37 + 11.5i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.90 - 3.21i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.154 - 0.213i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.173 - 0.126i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (-11.5 - 8.42i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.43604365037219678111691047781, −10.39084996033284074764949273703, −9.257706801397860057785029091739, −7.84000613862873866600441447472, −7.43260390856888252838344049019, −6.26805077786823553537155491393, −5.68556798040962540688410548656, −4.51511891344598598722340885195, −3.07225656533659394890612651058, −1.48862741382674839968098855818, 1.31578070255337729428422521631, 3.24823255795640406175791852464, 4.22281371948958440735017385889, 5.19579972216395749483671403062, 5.97349505133303288047942484095, 7.01361293681522816135754167815, 8.780558938075314023275326575193, 9.220036669346370257541978189034, 10.34783810923608034709410052202, 11.11148131894018196892588617408

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