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Properties

Label 2-1320-11.4-c1-0-1
Degree $2$
Conductor $1320$
Sign $-0.972 - 0.231i$
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 + (1.02 + 3.15i)7-s + (−0.809 − 0.587i)9-s + (−2.09 − 2.56i)11-s + (−2.68 − 1.94i)13-s + (0.309 + 0.951i)15-s + (−3.68 + 2.67i)17-s + (−1.92 + 5.92i)19-s − 3.31·21-s − 8.41·23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (1.40 + 4.31i)29-s + (7.58 + 5.51i)31-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (0.361 − 0.262i)5-s + (0.387 + 1.19i)7-s + (−0.269 − 0.195i)9-s + (−0.632 − 0.774i)11-s + (−0.743 − 0.540i)13-s + (0.0797 + 0.245i)15-s + (−0.894 + 0.649i)17-s + (−0.441 + 1.35i)19-s − 0.724·21-s − 1.75·23-s + (0.0618 − 0.190i)25-s + (0.155 − 0.113i)27-s + (0.260 + 0.801i)29-s + (1.36 + 0.990i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.972 - 0.231i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ -0.972 - 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6525070665\)
\(L(\frac12)\) \(\approx\) \(0.6525070665\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 + 2.56i)T \)
good7 \( 1 + (-1.02 - 3.15i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.68 + 1.94i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.68 - 2.67i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.92 - 5.92i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.41T + 23T^{2} \)
29 \( 1 + (-1.40 - 4.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.58 - 5.51i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.21 + 6.82i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.755 + 2.32i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.193T + 43T^{2} \)
47 \( 1 + (0.575 - 1.77i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.8 + 7.87i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.93 - 5.94i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (10.4 - 7.57i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.68T + 67T^{2} \)
71 \( 1 + (-7.41 + 5.39i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.53 - 10.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.248 + 0.180i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.84 - 3.51i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.45T + 89T^{2} \)
97 \( 1 + (3.01 + 2.19i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.19756462717980423433477369187, −9.114698235854619633076778077719, −8.425393461950206987474786167696, −7.934635204151460352566152074642, −6.38512881839782096658881257871, −5.73542426449794832678390896894, −5.12882219677787930953226246300, −4.09167415483177085000817896371, −2.85657160864940751768957673425, −1.88063268894196224388456383334, 0.25148305201767190907144625299, 1.89779515013713946894209652834, 2.71117678144990023882441419376, 4.47819740735725043000386442723, 4.68731040102706596345365062237, 6.20743689479600365606869049364, 6.82459459200185985330883118192, 7.56330197390539513445043625574, 8.178624208473149370371649255211, 9.486470986439140199919321107539

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