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Properties

Label 2-462-7.2-c1-0-2
Degree $2$
Conductor $462$
Sign $-0.0633 - 0.997i$
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 + (−1.32 + 2.29i)5-s + 0.999·6-s + (−1.32 + 2.29i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.32 + 2.29i)10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s − 4·13-s + (1.32 + 2.29i)14-s − 2.64·15-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.591 + 1.02i)5-s + 0.408·6-s + (−0.499 + 0.866i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.418 + 0.724i)10-s + (−0.150 − 0.261i)11-s + (0.144 − 0.249i)12-s − 1.10·13-s + (0.353 + 0.612i)14-s − 0.683·15-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.0633 - 0.997i$
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: \((2,\ 462,\ (\ :1/2),\ -0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.768485 + 0.818799i\)
\(L(\frac12)\) \(\approx\) \(0.768485 + 0.818799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 2.29i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.32 - 2.29i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.64 - 4.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 2.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.645 - 1.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 9.29T + 43T^{2} \)
47 \( 1 + (-1.96 + 3.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.29 + 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.96 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.79 + 6.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.70T + 71T^{2} \)
73 \( 1 + (-7.64 - 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.32 - 9.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + (1.35 - 2.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−11.16045834739989419786512282180, −10.46407327854974192556978961413, −9.755903455491698457253903407131, −8.729259109311995424705759140376, −7.72821984636365226556051338608, −6.50541822605150723395310920107, −5.51325688823133559166831762718, −4.23120689376415514267318213428, −3.20049014900062608103043681113, −2.42344807319156054735336464516, 0.58417469204313550461209308182, 2.78082280494689514864613226168, 4.23123868643961847590563129997, 4.89093028824853440245298174832, 6.25741544105369216520054575766, 7.44574993277929795428317435765, 7.64602945157985407868177621012, 8.951660518867071564001943695282, 9.582383009834323464837968917158, 10.92151123655604477143580594196

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