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Properties

Label 2-462-77.54-c1-0-15
Degree $2$
Conductor $462$
Sign $-0.0172 + 0.999i$
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 + (−2.13 − 1.23i)5-s − 0.999·6-s + (−0.941 − 2.47i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.23 − 2.13i)10-s + (−0.881 − 3.19i)11-s + (−0.866 − 0.499i)12-s − 1.32·13-s + (0.420 − 2.61i)14-s + 2.46·15-s + (−0.5 + 0.866i)16-s + (−2.23 − 3.87i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.953 − 0.550i)5-s − 0.408·6-s + (−0.356 − 0.934i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.389 − 0.674i)10-s + (−0.265 − 0.964i)11-s + (−0.249 − 0.144i)12-s − 0.366·13-s + (0.112 − 0.698i)14-s + 0.635·15-s + (−0.125 + 0.216i)16-s + (−0.542 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0172 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0172 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.0172 + 0.999i$
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: \((2,\ 462,\ (\ :1/2),\ -0.0172 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.590292 - 0.600558i\)
\(L(\frac12)\) \(\approx\) \(0.590292 - 0.600558i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.941 + 2.47i)T \)
11 \( 1 + (0.881 + 3.19i)T \)
good5 \( 1 + (2.13 + 1.23i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 + (2.23 + 3.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.21 - 3.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.14 + 7.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.44iT - 29T^{2} \)
31 \( 1 + (-2.34 + 1.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.46 + 4.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.18T + 41T^{2} \)
43 \( 1 - 9.63iT - 43T^{2} \)
47 \( 1 + (0.664 + 0.383i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.945 - 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.74 + 3.89i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 - 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.861 - 1.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (5.58 + 9.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.53 - 1.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (10.9 + 6.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.37iT - 97T^{2} \)
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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

−11.00293606026912183610164435155, −10.15403838318587575359506611578, −8.829177595440126934984695672025, −7.972126777278896999819346428353, −7.01494439723209863442796802940, −6.11876176433271420327330787621, −4.82588902092967270223366615936, −4.23096192289173228506585480731, −3.10048563281691787367844575750, −0.43803464919402534802219571642, 2.09020101125918315651790817300, 3.30770174027465785547916915744, 4.54231340537508420690208997478, 5.50268877540583681532408764047, 6.67470643132544328460271767887, 7.29826147492220896044811644991, 8.532431850935673027185655876271, 9.713179564048285095273962163536, 10.66262399120328261299645120297, 11.53755438141087419135200250920

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