Properties

Label 2-462-33.8-c1-0-18
Degree $2$
Conductor $462$
Sign $-0.443 + 0.896i$
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 + (−1.67 + 0.445i)3-s + (0.309 − 0.951i)4-s + (1.77 − 2.44i)5-s + (−1.09 + 1.34i)6-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + (2.60 − 1.49i)9-s − 3.02i·10-s + (−2.94 − 1.51i)11-s + (−0.0930 + 1.72i)12-s + (1.46 + 2.01i)13-s + (−0.951 + 0.309i)14-s + (−1.88 + 4.88i)15-s + (−0.809 − 0.587i)16-s + (−2.63 − 1.91i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.966 + 0.257i)3-s + (0.154 − 0.475i)4-s + (0.794 − 1.09i)5-s + (−0.445 + 0.548i)6-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + (0.867 − 0.497i)9-s − 0.955i·10-s + (−0.889 − 0.457i)11-s + (−0.0268 + 0.499i)12-s + (0.406 + 0.559i)13-s + (−0.254 + 0.0825i)14-s + (−0.486 + 1.26i)15-s + (−0.202 − 0.146i)16-s + (−0.640 − 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.443 + 0.896i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706968 - 1.13892i\)
\(L(\frac12)\) \(\approx\) \(0.706968 - 1.13892i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 0.445i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (2.94 + 1.51i)T \)
good5 \( 1 + (-1.77 + 2.44i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.46 - 2.01i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.63 + 1.91i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.06 - 0.671i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.69iT - 23T^{2} \)
29 \( 1 + (-2.83 + 8.71i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.14 + 3.73i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.36 + 4.21i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.00 - 9.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + (9.50 - 3.08i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.92 - 8.15i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.10 - 1.33i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.82 - 2.51i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (-6.23 + 8.57i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.37 + 1.74i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.33 + 1.83i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.58 - 5.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.7iT - 89T^{2} \)
97 \( 1 + (-10.8 + 7.87i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.86992561151836626854117499330, −9.986585536782417513624971453584, −9.314364546854584847658672455204, −8.150519758053704853433683132069, −6.42544007752752608563234594321, −6.02059388803830180108482019103, −4.83670141164226732339237386071, −4.30990708770602954055927274140, −2.44838153579977043109053718087, −0.77140389369839957222872762188, 2.12252282115464081919050881059, 3.43876959095025844886566097629, 5.02258520768342761282139121493, 5.73884380821749932232936731659, 6.67001806485769688447911129361, 7.14346626332954285539837010364, 8.424034745758781007673361605804, 9.960135803690426365603624956590, 10.52024534954088695405366352324, 11.26720671497436204673412189364

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