Properties

Label 2-462-77.54-c1-0-7
Degree $2$
Conductor $462$
Sign $0.999 + 0.00734i$
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 + (1.15 − 3.10i)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 + (0.348 − 0.937i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.999 + 0.00734i$
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: \((2,\ 462,\ (\ :1/2),\ 0.999 + 0.00734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27667 - 0.00468641i\)
\(L(\frac12)\) \(\approx\) \(1.27667 - 0.00468641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 + 3.10i)T \)
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

−10.68400885898747604770936118378, −10.37962737436967171764005792339, −9.573081385305957647243088710644, −8.533012729969139775317904069446, −7.29283280767693244577321515585, −6.33631244366734637454621357159, −5.65600118965301271555856060729, −4.08670368293715982402907668213, −2.71397765206563790399271145027, −1.35268240223151840511931003061, 1.38105287776608291993295119116, 2.27411970935197369960676325994, 5.08916467467300106275401209622, 5.20569032812160663857297737355, 6.37681988278186123261944268264, 7.28823520349332295810095712069, 8.594855662461470491418964640648, 9.301424990907745354524626896858, 9.821560045395215494336658766565, 10.93126536200374789772330745257

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