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Properties

Label 2-462-21.17-c1-0-0
Degree $2$
Conductor $462$
Sign $-0.832 - 0.553i$
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 + 1.5i)3-s + (0.499 − 0.866i)4-s + (−1.73 − 3i)5-s − 1.73i·6-s + (2 + 1.73i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (3 + 1.73i)10-s + (0.866 + 0.5i)11-s + (0.866 + 1.49i)12-s + 3.46i·13-s + (−2.59 − 0.499i)14-s + 6·15-s + (−0.5 − 0.866i)16-s + (−2.59 + 4.5i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 + 0.866i)3-s + (0.249 − 0.433i)4-s + (−0.774 − 1.34i)5-s − 0.707i·6-s + (0.755 + 0.654i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.948 + 0.547i)10-s + (0.261 + 0.150i)11-s + (0.250 + 0.433i)12-s + 0.960i·13-s + (−0.694 − 0.133i)14-s + 1.54·15-s + (−0.125 − 0.216i)16-s + (−0.630 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141040 + 0.466691i\)
\(L(\frac12)\) \(\approx\) \(0.141040 + 0.466691i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
7 \( 1 + (-2 - 1.73i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (1.73 + 3i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (2.59 - 4.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9iT - 71T^{2} \)
73 \( 1 + (-12 - 6.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (1.73 + 3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−11.22266480870327383841134750861, −10.63740985268530776188510429284, −9.233250031484436205070971890271, −8.807524855494675712760367786252, −8.212776418942662847009017605794, −6.77707389399582819479780444936, −5.63060052824502241178172134725, −4.73060323620743125784186751040, −4.00178718138852399964818011327, −1.61851435225085453909765467026, 0.39534873218681587455080037510, 2.22083774451300254987250689946, 3.40875978174255618571032629626, 4.88132077751938466840646066442, 6.48220591293010560033066362382, 7.12872611594949100899413718697, 7.78952331463571728781116953262, 8.615982662083283757920625933734, 10.22248733482494360473598965506, 10.84573630476882367113804250263

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