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Properties

Label 2-462-33.17-c1-0-21
Degree $2$
Conductor $462$
Sign $-0.483 + 0.875i$
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 + (0.157 − 1.72i)3-s + (−0.809 + 0.587i)4-s + (3.34 + 1.08i)5-s + (−1.68 + 0.382i)6-s + (−0.587 − 0.809i)7-s + (0.809 + 0.587i)8-s + (−2.95 − 0.544i)9-s − 3.51i·10-s + (−3.00 − 1.39i)11-s + (0.886 + 1.48i)12-s + (3.19 − 1.03i)13-s + (−0.587 + 0.809i)14-s + (2.40 − 5.59i)15-s + (0.309 − 0.951i)16-s + (1.68 − 5.18i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.0910 − 0.995i)3-s + (−0.404 + 0.293i)4-s + (1.49 + 0.485i)5-s + (−0.689 + 0.156i)6-s + (−0.222 − 0.305i)7-s + (0.286 + 0.207i)8-s + (−0.983 − 0.181i)9-s − 1.11i·10-s + (−0.906 − 0.421i)11-s + (0.255 + 0.429i)12-s + (0.885 − 0.287i)13-s + (−0.157 + 0.216i)14-s + (0.619 − 1.44i)15-s + (0.0772 − 0.237i)16-s + (0.408 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.744619 - 1.26137i\)
\(L(\frac12)\) \(\approx\) \(0.744619 - 1.26137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.157 + 1.72i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (3.00 + 1.39i)T \)
good5 \( 1 + (-3.34 - 1.08i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-3.19 + 1.03i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.68 + 5.18i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.42 + 4.70i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.40iT - 23T^{2} \)
29 \( 1 + (3.38 - 2.46i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.41 + 7.44i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.33 + 0.971i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.26 - 5.27i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.152iT - 43T^{2} \)
47 \( 1 + (5.17 - 7.12i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (11.1 - 3.60i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.54 - 3.50i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.90 - 1.91i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.03T + 67T^{2} \)
71 \( 1 + (-1.78 - 0.579i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.76 - 13.4i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.996 + 0.323i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.27 + 3.91i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 17.1iT - 89T^{2} \)
97 \( 1 + (2.50 + 7.69i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.04763445032561405093539042619, −9.690529463323927790420644086581, −9.307571627355956284503746701935, −7.972332287740681084012683070002, −7.14862045934863573868496417669, −6.01667735318730775674892398803, −5.29409371504379939471747419774, −3.16534283124004242683552863971, −2.46023038908215060475583781125, −1.04221566706911923812469040340, 1.90517204428348093454903891840, 3.59120222608955202426327556632, 5.00078295306585852694546271128, 5.67420760615916501922510469830, 6.34008628179135368956418230048, 8.006096389067685223809593089607, 8.752847357976281688291163250523, 9.594464539924820667630035439154, 10.14880273887703360181403290986, 10.87191629384211210049943425094

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