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Properties

Label 2-462-231.65-c1-0-15
Degree $2$
Conductor $462$
Sign $0.984 + 0.177i$
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 + (1.72 − 0.151i)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 + (2.95 − 0.522i)9-s + (0.246 − 0.142i)10-s + (2.18 − 2.49i)11-s + (−0.993 − 1.41i)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 + (0.996 − 0.0873i)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 + (0.984 − 0.174i)9-s + (0.0778 − 0.0449i)10-s + (0.658 − 0.752i)11-s + (−0.286 − 0.409i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52948 - 0.136903i\)
\(L(\frac12)\) \(\approx\) \(1.52948 - 0.136903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.151i)T \)
7 \( 1 + (2.09 + 1.61i)T \)
11 \( 1 + (-2.18 + 2.49i)T \)
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

−10.50837614937088276060447436601, −10.04874747370940920828860308933, −9.128169037496392740087000477169, −8.073219598910121980858162676924, −7.71336023271332179193221650195, −6.49116331905004475321208387916, −5.60140712745193448567030540675, −3.89183615327033280142961768017, −3.18843614088892958592623199603, −1.10131283291922488894924169478, 1.73599463759710753283953772182, 2.93469595304012076148748906050, 3.85884144370433383138417862416, 5.08738887419148964031587946581, 6.93616070345902435019750205764, 7.32941317121401092028866985767, 8.849065873772258144113724307076, 9.387929739503140447901472507724, 9.692817955434535518346852172588, 11.09912919205719702441184298957

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