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Properties

Label 2-1320-11.3-c1-0-3
Degree $2$
Conductor $1320$
Sign $-0.762 - 0.647i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
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  + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)5-s + (0.640 − 1.97i)7-s + (−0.809 + 0.587i)9-s + (−3.31 − 0.172i)11-s + (−1.56 + 1.13i)13-s + (0.309 − 0.951i)15-s + (−1.39 − 1.01i)17-s + (2.18 + 6.73i)19-s + 2.07·21-s − 0.938·23-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (−3.03 + 9.35i)29-s + (−3.62 + 2.63i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (−0.361 − 0.262i)5-s + (0.242 − 0.745i)7-s + (−0.269 + 0.195i)9-s + (−0.998 − 0.0520i)11-s + (−0.434 + 0.315i)13-s + (0.0797 − 0.245i)15-s + (−0.337 − 0.245i)17-s + (0.501 + 1.54i)19-s + 0.452·21-s − 0.195·23-s + (0.0618 + 0.190i)25-s + (−0.155 − 0.113i)27-s + (−0.564 + 1.73i)29-s + (−0.650 + 0.472i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.762 - 0.647i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ -0.762 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7006909413\)
\(L(\frac12)\) \(\approx\) \(0.7006909413\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (3.31 + 0.172i)T \)
good7 \( 1 + (-0.640 + 1.97i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.56 - 1.13i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.39 + 1.01i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.18 - 6.73i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.938T + 23T^{2} \)
29 \( 1 + (3.03 - 9.35i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.62 - 2.63i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.23 - 9.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.29 + 3.99i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.15T + 43T^{2} \)
47 \( 1 + (-1.41 - 4.36i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.98 - 2.89i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.10 + 12.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.38 + 3.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.27T + 67T^{2} \)
71 \( 1 + (3.55 + 2.58i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.959 + 2.95i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.90 - 4.29i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.80 - 4.22i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 + (8.66 - 6.29i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.01598954011990188881636320288, −9.181793524604071210248252659896, −8.254990502538665661377426182508, −7.65124103559231282259905348200, −6.84078014098894461769745517982, −5.49582804511377554916911656997, −4.87913350709961829984282131830, −3.91357543237354181153213740540, −3.07769137875803307792688609014, −1.58986633763859403031405000560, 0.27041700895469617569381605430, 2.19841788923155304353219027440, 2.77840650535898222628676875132, 4.11753768724253723569880037848, 5.25530951710436738058443956096, 5.91025351684443460289560786086, 7.12124662669175014549187311104, 7.59961629298619741843845007133, 8.450047020107204155697153551269, 9.185339111811228541180422829770

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