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Properties

Label 2-462-11.3-c1-0-11
Degree $2$
Conductor $462$
Sign $-0.958 - 0.285i$
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  + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−3.31 − 2.41i)5-s + (−0.809 − 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 4.10·10-s + (−1.81 + 2.77i)11-s − 12-s + (1.37 − 1.00i)13-s + (0.309 + 0.951i)14-s + (−1.26 + 3.90i)15-s + (−0.809 − 0.587i)16-s + (−1.5 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−1.48 − 1.07i)5-s + (−0.330 − 0.239i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s − 1.29·10-s + (−0.548 + 0.836i)11-s − 0.288·12-s + (0.381 − 0.277i)13-s + (0.0825 + 0.254i)14-s + (−0.327 + 1.00i)15-s + (−0.202 − 0.146i)16-s + (−0.363 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.958 - 0.285i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.958 - 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105318 + 0.721822i\)
\(L(\frac12)\) \(\approx\) \(0.105318 + 0.721822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (1.81 - 2.77i)T \)
good5 \( 1 + (3.31 + 2.41i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.37 + 1.00i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.5 + 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.901 + 2.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.57T + 23T^{2} \)
29 \( 1 + (0.0570 - 0.175i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.95 + 3.59i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.03 + 9.32i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.74 + 8.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + (-0.560 - 1.72i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.94 + 5.04i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.94 + 5.97i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.11 + 5.16i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.31T + 67T^{2} \)
71 \( 1 + (-8.87 - 6.44i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.32 - 10.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.98 + 7.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.87 + 1.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.52T + 89T^{2} \)
97 \( 1 + (-0.390 + 0.284i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.94429739377107964693373852717, −9.710834740727897927775694713564, −8.563418503779008481999539640562, −7.87354104720902046843339568284, −6.89192614698852817998728241568, −5.55724109275054357021301070448, −4.65523235848769391349147914385, −3.75777419399478695659649580483, −2.17719215864605917830569927904, −0.36928247793994106862857276834, 3.00484551042550051837048390657, 3.77594215115209146614374748331, 4.59077862144866316110032686452, 6.09289619931247140315025446101, 6.76012032202645448474304567749, 8.004849177625132332242772144849, 8.334811681547051843975611499298, 10.10667312852659769119434289333, 10.74486698898958304311276349751, 11.60919689103790658027528370841

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