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Properties

Label 2-1320-11.4-c1-0-4
Degree $2$
Conductor $1320$
Sign $-0.796 - 0.604i$
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 + (1.50 + 4.64i)7-s + (−0.809 − 0.587i)9-s + (0.881 + 3.19i)11-s + (2.56 + 1.86i)13-s + (−0.309 − 0.951i)15-s + (1.69 − 1.23i)17-s + (−0.0706 + 0.217i)19-s − 4.88·21-s − 2.62·23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−1.67 − 5.14i)29-s + (1.23 + 0.899i)31-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (−0.361 + 0.262i)5-s + (0.570 + 1.75i)7-s + (−0.269 − 0.195i)9-s + (0.265 + 0.964i)11-s + (0.712 + 0.517i)13-s + (−0.0797 − 0.245i)15-s + (0.411 − 0.298i)17-s + (−0.0162 + 0.0498i)19-s − 1.06·21-s − 0.547·23-s + (0.0618 − 0.190i)25-s + (0.155 − 0.113i)27-s + (−0.310 − 0.955i)29-s + (0.222 + 0.161i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.796 - 0.604i$
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: \((2,\ 1320,\ (\ :1/2),\ -0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.392994042\)
\(L(\frac12)\) \(\approx\) \(1.392994042\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.881 - 3.19i)T \)
good7 \( 1 + (-1.50 - 4.64i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.56 - 1.86i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.69 + 1.23i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0706 - 0.217i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 + (1.67 + 5.14i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.23 - 0.899i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.370 + 1.14i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.0371 - 0.114i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.02T + 43T^{2} \)
47 \( 1 + (3.27 - 10.0i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.82 + 1.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.41 + 10.5i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.83 - 4.96i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.63T + 67T^{2} \)
71 \( 1 + (5.10 - 3.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.37 - 7.30i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.45 - 6.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.5 + 9.09i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.53T + 89T^{2} \)
97 \( 1 + (2.29 + 1.66i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.709701964222877689522542796864, −9.279390293840836577022976925227, −8.397243378074037080067206587337, −7.69326593717660872419000131995, −6.46328470793275991514963609787, −5.77978907374959903288529677454, −4.86608270658155703156897905194, −4.05575958269398192537870147617, −2.82763444504102381954429697002, −1.80488672070414371984633956927, 0.63272266788671106537660244513, 1.49127695983512885526562578334, 3.34231670170983202236229531169, 4.00618599923307541636638194886, 5.08138014627472278031944256793, 6.08872116240420997815921423540, 6.93491328124854950020301687536, 7.82193521605699278515610590258, 8.180355043425578705645967243705, 9.191187246946824364312788606608

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