Properties

Label 2-462-33.2-c1-0-6
Degree $2$
Conductor $462$
Sign $0.510 - 0.859i$
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  + (0.309 − 0.951i)2-s + (0.391 + 1.68i)3-s + (−0.809 − 0.587i)4-s + (−0.599 + 0.194i)5-s + (1.72 + 0.148i)6-s + (−0.587 + 0.809i)7-s + (−0.809 + 0.587i)8-s + (−2.69 + 1.32i)9-s + 0.630i·10-s + (2.44 + 2.24i)11-s + (0.674 − 1.59i)12-s + (2.41 + 0.783i)13-s + (0.587 + 0.809i)14-s + (−0.563 − 0.935i)15-s + (0.309 + 0.951i)16-s + (1.70 + 5.24i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.226 + 0.974i)3-s + (−0.404 − 0.293i)4-s + (−0.268 + 0.0871i)5-s + (0.704 + 0.0607i)6-s + (−0.222 + 0.305i)7-s + (−0.286 + 0.207i)8-s + (−0.897 + 0.440i)9-s + 0.199i·10-s + (0.737 + 0.675i)11-s + (0.194 − 0.460i)12-s + (0.669 + 0.217i)13-s + (0.157 + 0.216i)14-s + (−0.145 − 0.241i)15-s + (0.0772 + 0.237i)16-s + (0.413 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.510 - 0.859i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.510 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22278 + 0.696030i\)
\(L(\frac12)\) \(\approx\) \(1.22278 + 0.696030i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.391 - 1.68i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (-2.44 - 2.24i)T \)
good5 \( 1 + (0.599 - 0.194i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-2.41 - 0.783i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.70 - 5.24i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.782 + 1.07i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.39iT - 23T^{2} \)
29 \( 1 + (-0.707 - 0.513i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.94 + 9.06i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.24 + 0.908i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.18 - 2.31i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.14iT - 43T^{2} \)
47 \( 1 + (1.24 + 1.71i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.81 - 0.915i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.03 + 5.54i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-11.4 + 3.73i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.01T + 67T^{2} \)
71 \( 1 + (-13.8 + 4.51i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.66 + 2.28i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (9.35 + 3.03i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.56 + 10.9i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 + (-1.98 + 6.11i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.38117138991137708374921280076, −10.19982074345405741186551665168, −9.610317798011663612448290434236, −8.803636769469381676845631824832, −7.80425174237950223934495994940, −6.26405292482496569773757013730, −5.30705622029303921221806754808, −4.00306812491219810036711427685, −3.54743057159913932088874521839, −1.93598970719606996764048085873, 0.838890720635855906627407849613, 2.87531586823685707476877084791, 4.01380162597935425763787671396, 5.44231545218784031434506243774, 6.51669506937352755302255039086, 6.99262479973567083549222455267, 8.285516557176662117331695573234, 8.570472968277987778891967248860, 9.838078303357214937220091755312, 11.11484937764644286007563227153

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