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Properties

Label 2-1320-11.3-c1-0-2
Degree $2$
Conductor $1320$
Sign $-0.744 - 0.667i$
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 + (−1.06 + 3.27i)7-s + (−0.809 + 0.587i)9-s + (3.30 + 0.260i)11-s + (−3.64 + 2.64i)13-s + (0.309 − 0.951i)15-s + (−4.13 − 3.00i)17-s + (0.166 + 0.513i)19-s + 3.43·21-s − 8.15·23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−0.666 + 2.05i)29-s + (−2.99 + 2.17i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.361 + 0.262i)5-s + (−0.401 + 1.23i)7-s + (−0.269 + 0.195i)9-s + (0.996 + 0.0786i)11-s + (−1.01 + 0.733i)13-s + (0.0797 − 0.245i)15-s + (−1.00 − 0.728i)17-s + (0.0382 + 0.117i)19-s + 0.750·21-s − 1.70·23-s + (0.0618 + 0.190i)25-s + (0.155 + 0.113i)27-s + (−0.123 + 0.381i)29-s + (−0.538 + 0.391i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.744 - 0.667i$
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: \((2,\ 1320,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6113516267\)
\(L(\frac12)\) \(\approx\) \(0.6113516267\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 - 0.260i)T \)
good7 \( 1 + (1.06 - 3.27i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.64 - 2.64i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.13 + 3.00i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.166 - 0.513i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.15T + 23T^{2} \)
29 \( 1 + (0.666 - 2.05i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.99 - 2.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.40 + 4.31i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.94 + 12.1i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.93T + 43T^{2} \)
47 \( 1 + (-2.26 - 6.95i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.860 - 0.624i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.46 - 7.59i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.36 + 2.44i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.39T + 67T^{2} \)
71 \( 1 + (-11.7 - 8.51i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.163 + 0.501i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (14.3 - 10.4i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.59 - 6.24i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + (5.64 - 4.10i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.661251752566519994283873292348, −9.228768692576515713291104969097, −8.469484232023979252861824393823, −7.25149329006176687227994344423, −6.67073231641079445310048965969, −5.92268655614330091237717053396, −5.07278062161361913547075626556, −3.87503920056257317708140974135, −2.50085734391799989530863857595, −1.89107132721615599352175453371, 0.23847059556060094268505007256, 1.84280881418743897569865222364, 3.36469418379286581723344213968, 4.17474230754166087450865153619, 4.91144304138433030100409289064, 6.13111650041337058593580994173, 6.67264025546296065498989783753, 7.75216076225028813985576411007, 8.545370817157290561415041654139, 9.716808640174682460284598982710

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