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Properties

Label 2-462-77.13-c1-0-13
Degree $2$
Conductor $462$
Sign $0.698 + 0.715i$
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  + (0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (0.169 − 0.0550i)5-s + (−0.309 − 0.951i)6-s + (−0.364 − 2.62i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + 0.178·10-s + (1.87 − 2.73i)11-s − 0.999i·12-s + (0.936 − 2.88i)13-s + (0.462 − 2.60i)14-s + (−0.144 − 0.104i)15-s + (0.309 + 0.951i)16-s + (0.531 + 1.63i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (−0.339 − 0.467i)3-s + (0.404 + 0.293i)4-s + (0.0757 − 0.0246i)5-s + (−0.126 − 0.388i)6-s + (−0.137 − 0.990i)7-s + (0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + 0.0563·10-s + (0.564 − 0.825i)11-s − 0.288i·12-s + (0.259 − 0.799i)13-s + (0.123 − 0.696i)14-s + (−0.0372 − 0.0270i)15-s + (0.0772 + 0.237i)16-s + (0.128 + 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.698 + 0.715i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74276 - 0.734682i\)
\(L(\frac12)\) \(\approx\) \(1.74276 - 0.734682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.364 + 2.62i)T \)
11 \( 1 + (-1.87 + 2.73i)T \)
good5 \( 1 + (-0.169 + 0.0550i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.936 + 2.88i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.531 - 1.63i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.99 + 2.17i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 + (-1.15 + 1.58i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.141 + 0.0460i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.77 + 2.74i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.410 - 0.298i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.47iT - 43T^{2} \)
47 \( 1 + (-5.74 - 7.90i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.248 + 0.764i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.274 - 0.377i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.45 - 13.7i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + (-0.790 - 2.43i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.70 + 5.59i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.257 + 0.0837i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.03 - 6.26i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 17.2iT - 89T^{2} \)
97 \( 1 + (4.38 + 1.42i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.08195685931300925141112597407, −10.33913136438345503117886615064, −9.051807646077228770648888525505, −7.88093019494056392805007392945, −7.15904584650044984923013074310, −6.18397409936175758276538574887, −5.38160022695124295800862035654, −4.07769199697014397591570891310, −3.05908940485012577751132086338, −1.10976048211512892694604888314, 1.88914198319642726985141201721, 3.30121859386549251257820595855, 4.45150689943484990204926687839, 5.36634240494874278398493389380, 6.28408088981043118192868678583, 7.21054939046366887448086734981, 8.706750591524665518502675565158, 9.530844876270970201207761752535, 10.27007882629837592400245270714, 11.51132212115488210886323523631

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