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Properties

Label 2-1320-11.3-c1-0-20
Degree $2$
Conductor $1320$
Sign $-0.631 + 0.775i$
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 + (−0.591 + 1.81i)7-s + (−0.809 + 0.587i)9-s + (0.105 − 3.31i)11-s + (−2.22 + 1.61i)13-s + (0.309 − 0.951i)15-s + (−2.58 − 1.87i)17-s + (−0.756 − 2.32i)19-s − 1.91·21-s − 7.03·23-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.0488 − 0.150i)29-s + (3.62 − 2.63i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (−0.361 − 0.262i)5-s + (−0.223 + 0.687i)7-s + (−0.269 + 0.195i)9-s + (0.0317 − 0.999i)11-s + (−0.615 + 0.447i)13-s + (0.0797 − 0.245i)15-s + (−0.626 − 0.454i)17-s + (−0.173 − 0.534i)19-s − 0.417·21-s − 1.46·23-s + (0.0618 + 0.190i)25-s + (−0.155 − 0.113i)27-s + (0.00907 − 0.0279i)29-s + (0.650 − 0.472i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.631 + 0.775i$
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: \((2,\ 1320,\ (\ :1/2),\ -0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3201384950\)
\(L(\frac12)\) \(\approx\) \(0.3201384950\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.105 + 3.31i)T \)
good7 \( 1 + (0.591 - 1.81i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.22 - 1.61i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.58 + 1.87i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.756 + 2.32i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.03T + 23T^{2} \)
29 \( 1 + (-0.0488 + 0.150i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.62 + 2.63i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.179 + 0.552i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.608 - 1.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.80T + 43T^{2} \)
47 \( 1 + (3.69 + 11.3i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.89 - 2.10i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.49 + 10.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.77 + 4.91i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + (-2.55 - 1.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.00 + 3.08i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.8 - 7.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.76 - 4.18i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.802T + 89T^{2} \)
97 \( 1 + (-2.74 + 1.99i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.324562769369560515777029815195, −8.579809882319711882368041373291, −7.989818052275089347492881156136, −6.80257981428926921483936389723, −5.95870220124229514266377176672, −5.02230680266399225229123629740, −4.19696109211062065678240450783, −3.16794730806859489950383055457, −2.16270067279338663330044161245, −0.12199639061227559820846281947, 1.62095490324193133223514026371, 2.75048221363886550599157221587, 3.91157227699265886045377482119, 4.67560209628015717875318957729, 6.00017358337798838896090752946, 6.76137343502900959512718842384, 7.54356085991108876586759666163, 8.046791187374849066240803083391, 9.077027451461724922541257476034, 10.18731048692780927925190596672

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