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Properties

Label 2-462-77.54-c1-0-2
Degree $2$
Conductor $462$
Sign $-0.180 - 0.983i$
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 + (3.72 + 2.15i)5-s − 0.999·6-s + (−1.43 + 2.22i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (2.15 + 3.72i)10-s + (−3.27 − 0.553i)11-s + (−0.866 − 0.499i)12-s + 1.00·13-s + (−2.35 + 1.21i)14-s − 4.30·15-s + (−0.5 + 0.866i)16-s + (−1.66 − 2.88i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (1.66 + 0.961i)5-s − 0.408·6-s + (−0.541 + 0.840i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.680 + 1.17i)10-s + (−0.985 − 0.166i)11-s + (−0.249 − 0.144i)12-s + 0.277·13-s + (−0.628 + 0.323i)14-s − 1.11·15-s + (−0.125 + 0.216i)16-s + (−0.404 − 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.180 - 0.983i$
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: \((2,\ 462,\ (\ :1/2),\ -0.180 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26857 + 1.52186i\)
\(L(\frac12)\) \(\approx\) \(1.26857 + 1.52186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 - 2.22i)T \)
11 \( 1 + (3.27 + 0.553i)T \)
good5 \( 1 + (-3.72 - 2.15i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.00T + 13T^{2} \)
17 \( 1 + (1.66 + 2.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.61 + 2.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.86 + 4.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.74iT - 29T^{2} \)
31 \( 1 + (2.77 - 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 2.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.45T + 41T^{2} \)
43 \( 1 + 4.14iT - 43T^{2} \)
47 \( 1 + (-2.77 - 1.60i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.65 - 9.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.98 + 1.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.42 + 12.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.165 - 0.286i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.84T + 71T^{2} \)
73 \( 1 + (7.44 + 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.9 - 8.04i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + (2.41 + 1.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−11.03157060517410472022155147547, −10.62249738777075767493545391095, −9.536530358205605888663484331008, −8.845392110549615699224830832640, −7.10679707626689765449737713997, −6.49611376831689355533580934715, −5.57609057297829348882989956110, −5.08216990957357263411587484865, −3.12315802861782027370430773479, −2.38301425617002228889269251651, 1.15470396450481984446168800753, 2.37227829625718092234294480479, 4.07110040036181348121289915946, 5.27147478881339649717469536019, 5.82198671967374660498498760293, 6.74008133590502370217159586195, 8.022255120858065938167958262409, 9.379735961882063028577927754160, 10.06128066895805694261442902231, 10.66274680303681305396086519201

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