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Properties

Label 2-462-33.17-c1-0-23
Degree $2$
Conductor $462$
Sign $-0.788 + 0.615i$
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 + (1.05 − 1.37i)3-s + (−0.809 + 0.587i)4-s + (−3.96 − 1.28i)5-s + (1.63 + 0.574i)6-s + (0.587 + 0.809i)7-s + (−0.809 − 0.587i)8-s + (−0.789 − 2.89i)9-s − 4.16i·10-s + (−3.18 − 0.910i)11-s + (−0.0414 + 1.73i)12-s + (−4.46 + 1.45i)13-s + (−0.587 + 0.809i)14-s + (−5.93 + 4.10i)15-s + (0.309 − 0.951i)16-s + (−0.988 + 3.04i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.606 − 0.794i)3-s + (−0.404 + 0.293i)4-s + (−1.77 − 0.575i)5-s + (0.667 + 0.234i)6-s + (0.222 + 0.305i)7-s + (−0.286 − 0.207i)8-s + (−0.263 − 0.964i)9-s − 1.31i·10-s + (−0.961 − 0.274i)11-s + (−0.0119 + 0.499i)12-s + (−1.23 + 0.402i)13-s + (−0.157 + 0.216i)14-s + (−1.53 + 1.05i)15-s + (0.0772 − 0.237i)16-s + (−0.239 + 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.788 + 0.615i$
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: \((2,\ 462,\ (\ :1/2),\ -0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129894 - 0.377746i\)
\(L(\frac12)\) \(\approx\) \(0.129894 - 0.377746i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 + 1.37i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (3.18 + 0.910i)T \)
good5 \( 1 + (3.96 + 1.28i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (4.46 - 1.45i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.988 - 3.04i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.33 - 1.83i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 8.04iT - 23T^{2} \)
29 \( 1 + (0.464 - 0.337i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.60 + 8.00i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.79 + 4.93i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.287 - 0.209i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.99iT - 43T^{2} \)
47 \( 1 + (-4.03 + 5.55i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.78 - 1.55i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.33 + 3.21i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.60 + 2.47i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.12T + 67T^{2} \)
71 \( 1 + (-2.88 - 0.936i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.59 + 6.31i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.592 + 0.192i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.61 - 8.06i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 5.10iT - 89T^{2} \)
97 \( 1 + (0.238 + 0.733i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.94171815144873442864064938818, −9.365073587280899613440989693730, −8.402978688160030959504132885893, −7.932452953839638070217164631081, −7.34963069618026556955905553751, −6.17407169446808622109162863651, −4.77701955771487899929098118013, −3.95031519474188922593437527406, −2.55447226821842852629775898645, −0.20324180322761598255106631133, 2.65578848476108464034984132174, 3.40162361722697526374392975067, 4.47910066876424568210540715246, 5.11908912698730631666519937398, 7.34641429016855778582739997575, 7.66536165060963554625888289781, 8.763636753519113975154664250043, 9.855415474474922527224119128532, 10.63742236071506952465300674462, 11.27022832758081873416352610655

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