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Properties

Label 2-462-231.32-c1-0-27
Degree $2$
Conductor $462$
Sign $-0.999 - 0.0377i$
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 + (−0.993 − 1.41i)3-s + (−0.499 + 0.866i)4-s + (0.246 − 0.142i)5-s + (−0.731 + 1.56i)6-s + (2.09 − 1.61i)7-s + 0.999·8-s + (−1.02 + 2.81i)9-s + (−0.246 − 0.142i)10-s + (−3.25 − 0.644i)11-s + (1.72 − 0.151i)12-s − 6.02i·13-s + (−2.44 − 1.00i)14-s + (−0.446 − 0.207i)15-s + (−0.5 − 0.866i)16-s + (3.12 − 5.41i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.573 − 0.819i)3-s + (−0.249 + 0.433i)4-s + (0.110 − 0.0635i)5-s + (−0.298 + 0.640i)6-s + (0.791 − 0.611i)7-s + 0.353·8-s + (−0.341 + 0.939i)9-s + (−0.0778 − 0.0449i)10-s + (−0.980 − 0.194i)11-s + (0.498 − 0.0436i)12-s − 1.67i·13-s + (−0.654 − 0.268i)14-s + (−0.115 − 0.0536i)15-s + (−0.125 − 0.216i)16-s + (0.758 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.999 - 0.0377i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.999 - 0.0377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0131473 + 0.695892i\)
\(L(\frac12)\) \(\approx\) \(0.0131473 + 0.695892i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.993 + 1.41i)T \)
7 \( 1 + (-2.09 + 1.61i)T \)
11 \( 1 + (3.25 + 0.644i)T \)
good5 \( 1 + (-0.246 + 0.142i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 6.02iT - 13T^{2} \)
17 \( 1 + (-3.12 + 5.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.70 - 2.71i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.55 - 0.898i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.434T + 29T^{2} \)
31 \( 1 + (3.64 - 6.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.57 + 9.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.326T + 41T^{2} \)
43 \( 1 + 2.29iT - 43T^{2} \)
47 \( 1 + (3.52 - 2.03i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.66 + 0.960i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.38 - 4.83i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.37 + 1.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.34 + 7.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.88iT - 71T^{2} \)
73 \( 1 + (3.80 + 2.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.63 - 2.67i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.61T + 83T^{2} \)
89 \( 1 + (-14.8 + 8.56i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.98T + 97T^{2} \)
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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

−10.64239587957790732525226057884, −10.13689469410259831649475298076, −8.596836249217102219180664479342, −7.76773312020400379218377289118, −7.30342017612764920223930215176, −5.63954718608426466113152495602, −5.04181242367018230138519554493, −3.32605269636170541690863685566, −1.94600723104372112865737687126, −0.50880128092280259323201812068, 2.05850714934615216244732668404, 4.09436849443429866929723705529, 4.91861490853694798036493989789, 5.89904317451394198372307190624, 6.68853908133216420708463022455, 8.105783516617589501991764471027, 8.723017964101259559868726140086, 9.762891590094673155338829561228, 10.47037458600287754595270159976, 11.35287297207241355731974150977

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