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Properties

Label 2-462-77.54-c1-0-13
Degree $2$
Conductor $462$
Sign $0.905 + 0.425i$
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 + (−2.01 − 1.16i)5-s + 0.999·6-s + (1.31 − 2.29i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.16 − 2.01i)10-s + (2.46 − 2.21i)11-s + (0.866 + 0.499i)12-s + 1.44·13-s + (2.28 − 1.32i)14-s − 2.32·15-s + (−0.5 + 0.866i)16-s + (1.60 + 2.78i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.901 − 0.520i)5-s + 0.408·6-s + (0.497 − 0.867i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.367 − 0.637i)10-s + (0.743 − 0.668i)11-s + (0.249 + 0.144i)12-s + 0.399·13-s + (0.611 − 0.355i)14-s − 0.600·15-s + (−0.125 + 0.216i)16-s + (0.389 + 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.905 + 0.425i$
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: \((2,\ 462,\ (\ :1/2),\ 0.905 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12243 - 0.473465i\)
\(L(\frac12)\) \(\approx\) \(2.12243 - 0.473465i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 2.29i)T \)
11 \( 1 + (-2.46 + 2.21i)T \)
good5 \( 1 + (2.01 + 1.16i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.44T + 13T^{2} \)
17 \( 1 + (-1.60 - 2.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.07 + 5.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.14 - 5.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.26iT - 29T^{2} \)
31 \( 1 + (-2.67 + 1.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.57 - 7.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 - 1.89iT - 43T^{2} \)
47 \( 1 + (9.22 + 5.32i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.60 - 7.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.461 - 0.266i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.233 + 0.404i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.61 - 7.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.25T + 71T^{2} \)
73 \( 1 + (-2.50 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.21 - 4.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.56T + 83T^{2} \)
89 \( 1 + (3.79 + 2.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.8iT - 97T^{2} \)
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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

−11.48970688004812187178797936309, −10.12133406456081338848633224552, −8.796650946620499053980647076503, −8.151710697231345059762691731134, −7.38274434663981967734049180155, −6.45452928472276066060358529603, −5.10224159557474015642416504711, −4.03497609907603146582589451583, −3.35146186828858404039753899816, −1.26017861867522612164791279033, 1.94226760662342207976657616790, 3.25696873925040851708840076829, 4.09571897472435468980949918479, 5.17985648086171467404978222563, 6.38553779158584262290582273630, 7.53432480825166404206507333057, 8.346201076735388527029828203033, 9.438119643118128822904649400233, 10.29377861792439090828975679838, 11.36279876528378937977869112365

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