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Properties

Label 2-462-11.4-c1-0-9
Degree $2$
Conductor $462$
Sign $0.999 + 0.00892i$
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 + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (2.26 − 1.64i)5-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 2.80·10-s + (0.921 + 3.18i)11-s + 12-s + (−0.706 − 0.513i)13-s + (−0.309 + 0.951i)14-s + (−0.866 − 2.66i)15-s + (−0.809 + 0.587i)16-s + (3.92 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (1.01 − 0.736i)5-s + (0.330 − 0.239i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.886·10-s + (0.277 + 0.960i)11-s + 0.288·12-s + (−0.195 − 0.142i)13-s + (−0.0825 + 0.254i)14-s + (−0.223 − 0.688i)15-s + (−0.202 + 0.146i)16-s + (0.950 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.999 + 0.00892i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.999 + 0.00892i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36537 - 0.0105575i\)
\(L(\frac12)\) \(\approx\) \(2.36537 - 0.0105575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.921 - 3.18i)T \)
good5 \( 1 + (-2.26 + 1.64i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (0.706 + 0.513i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.92 + 2.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.10 + 3.40i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.89T + 23T^{2} \)
29 \( 1 + (0.687 + 2.11i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.70 + 3.41i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.83 - 5.64i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.18 - 6.73i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.74T + 43T^{2} \)
47 \( 1 + (3.53 - 10.8i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.1 + 7.34i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.04 - 6.29i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.35 + 6.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.98T + 67T^{2} \)
71 \( 1 + (8.26 - 6.00i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.85 + 14.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.97 - 3.61i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.08 - 5.87i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-2.34 - 1.70i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.38979414661959487017094302365, −9.762101280980309229878164432576, −9.367560028024718472374536541898, −8.155155402317172843019761677469, −7.32071052722940454999439261103, −6.25559025236054907048251260027, −5.40600534954254483460247002759, −4.55504068662840320934763850438, −2.85492963155460446983182374479, −1.62523047471475921994580916532, 1.77876089155890585001076755936, 3.15761926783736901780030235946, 3.96342461850047429372857781065, 5.49260081524061660851173109417, 6.00458123527762679501117238702, 7.22178050616432882794529007646, 8.509472322009598462560868159901, 9.604184478787588452681722735995, 10.33013087246871156958783418315, 10.82552650837098329070684655135

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