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Properties

Label 2-462-77.10-c1-0-12
Degree $2$
Conductor $462$
Sign $-0.655 + 0.755i$
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 + (−2.01 + 1.16i)5-s − 0.999·6-s + (−1.31 − 2.29i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.16 − 2.01i)10-s + (−3.15 − 1.02i)11-s + (0.866 − 0.499i)12-s − 1.44·13-s + (2.28 + 1.32i)14-s − 2.32·15-s + (−0.5 − 0.866i)16-s + (−1.60 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.901 + 0.520i)5-s − 0.408·6-s + (−0.497 − 0.867i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.367 − 0.637i)10-s + (−0.950 − 0.309i)11-s + (0.249 − 0.144i)12-s − 0.399·13-s + (0.611 + 0.355i)14-s − 0.600·15-s + (−0.125 − 0.216i)16-s + (−0.389 + 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.655 + 0.755i$
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: \((2,\ 462,\ (\ :1/2),\ -0.655 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0589172 - 0.129149i\)
\(L(\frac12)\) \(\approx\) \(0.0589172 - 0.129149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 2.29i)T \)
11 \( 1 + (3.15 + 1.02i)T \)
good5 \( 1 + (2.01 - 1.16i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 + (1.60 - 2.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.07 + 5.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.14 + 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.26iT - 29T^{2} \)
31 \( 1 + (-2.67 - 1.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.57 + 7.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 - 1.89iT - 43T^{2} \)
47 \( 1 + (9.22 - 5.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.60 + 7.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.461 + 0.266i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.233 + 0.404i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.61 + 7.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.25T + 71T^{2} \)
73 \( 1 + (2.50 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.21 - 4.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.56T + 83T^{2} \)
89 \( 1 + (3.79 - 2.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−10.70055953255137140832640896350, −9.892539585955318004085407804197, −8.760917337727687873441419045474, −7.999023940577895368151598590532, −7.21138204966387577118688195021, −6.42365854881285139261982131004, −4.81454369852318269539971545329, −3.73279680002246730978888542785, −2.53841176807566674272546591103, −0.093167794247789739111031788439, 2.04465035542777424217577258005, 3.17667927421112547337843652843, 4.42734230460183777559250573000, 5.80839148143442391316748956264, 7.13200735614889836452681574154, 8.053310945528534307872751380550, 8.497034207809835620733457809074, 9.598730694641361341050541004558, 10.21916796098681614556908126398, 11.65710112414858495830345524617

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