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Properties

Label 2-462-77.54-c1-0-4
Degree $2$
Conductor $462$
Sign $0.306 - 0.951i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.725 + 0.418i)5-s − 0.999·6-s + (2.44 + 1.02i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.418 + 0.725i)10-s + (2.45 − 2.22i)11-s + (−0.866 − 0.499i)12-s − 2.59·13-s + (1.60 + 2.10i)14-s − 0.837·15-s + (−0.5 + 0.866i)16-s + (2.98 + 5.17i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.324 + 0.187i)5-s − 0.408·6-s + (0.922 + 0.386i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.132 + 0.229i)10-s + (0.741 − 0.670i)11-s + (−0.249 − 0.144i)12-s − 0.719·13-s + (0.428 + 0.562i)14-s − 0.216·15-s + (−0.125 + 0.216i)16-s + (0.724 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.306 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56742 + 1.14189i\)
\(L(\frac12)\) \(\approx\) \(1.56742 + 1.14189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.44 - 1.02i)T \)
11 \( 1 + (-2.45 + 2.22i)T \)
good5 \( 1 + (-0.725 - 0.418i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.59T + 13T^{2} \)
17 \( 1 + (-2.98 - 5.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.55 - 2.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.43 - 2.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.38iT - 29T^{2} \)
31 \( 1 + (0.913 - 0.527i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.49 + 9.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 1.27iT - 43T^{2} \)
47 \( 1 + (10.6 + 6.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.58 + 4.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.38 - 4.84i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.03 + 3.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.51 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (4.95 + 8.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (11.7 + 6.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.99T + 83T^{2} \)
89 \( 1 + (-7.28 - 4.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.786iT - 97T^{2} \)
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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

−11.28137773041310755513976620606, −10.56479082779166383218809773277, −9.461606857768660366405106443114, −8.374476977510867936466599022135, −7.51896918246085687039197636549, −6.16669765006712712036832407351, −5.71635855989185920171823146257, −4.57914314895432267115760475347, −3.55498297095056633872746056484, −1.85866897173974893527635867886, 1.23320182942830634192487966893, 2.55486941381222994980286187477, 4.35273086023382463965386997970, 4.91989696089540567693745559581, 6.03673253527668138070022940474, 7.10431778815556685930897780059, 7.88284206456802269797108887597, 9.418570586526654782379077978015, 10.01244396240604651950774396499, 11.28635072403398923742471688748

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