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Properties

Label 2-462-7.2-c1-0-8
Degree $2$
Conductor $462$
Sign $-0.0627 + 0.998i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.60 − 2.77i)5-s + 0.999·6-s + (1.60 − 2.10i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.60 + 2.77i)10-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s − 6.40·13-s + (1.02 + 2.43i)14-s − 3.20·15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.715 − 1.23i)5-s + 0.408·6-s + (0.605 − 0.796i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.506 + 0.876i)10-s + (−0.150 − 0.261i)11-s + (−0.144 + 0.249i)12-s − 1.77·13-s + (0.273 + 0.652i)14-s − 0.826·15-s + (−0.125 + 0.216i)16-s + (−0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.0627 + 0.998i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.0627 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681421 - 0.725630i\)
\(L(\frac12)\) \(\approx\) \(0.681421 - 0.725630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.60 + 2.10i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.60 + 2.77i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 6.40T + 13T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.576 - 0.998i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.976 - 1.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.24T + 29T^{2} \)
31 \( 1 + (5.24 + 9.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.57 - 4.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.24T + 41T^{2} \)
43 \( 1 - 5.15T + 43T^{2} \)
47 \( 1 + (-3.02 + 5.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.20 - 5.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.77 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.85 - 8.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.12 + 5.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.24T + 71T^{2} \)
73 \( 1 + (-1.04 - 1.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.80 + 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.55T + 83T^{2} \)
89 \( 1 + (-9.20 + 15.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.49T + 97T^{2} \)
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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

−10.62884288740170453330715312645, −9.764023695548913610950092256202, −8.984389736512072204143669410901, −7.890095747268079473363697692419, −7.35718171193031065182537725431, −6.09767080909783209629805848043, −5.15885838274558759677279372969, −4.48460393377776174844978349390, −2.08791356632927779332887819520, −0.70621898919584185543757569775, 2.18584371259481185497844531597, 2.89214629357818907268672379455, 4.54699400551725565503628389554, 5.49723374202152058128231518776, 6.69626052770293619285429797374, 7.64893397346993051591994586481, 8.934592103472758290172396390393, 9.673873574026709927015010202508, 10.53871775811539621860987218295, 10.92223799806305380025328008573

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