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Properties

Label 2-1320-11.4-c1-0-11
Degree $2$
Conductor $1320$
Sign $0.739 + 0.673i$
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 + (−0.725 − 2.23i)7-s + (−0.809 − 0.587i)9-s + (0.594 + 3.26i)11-s + (3.20 + 2.32i)13-s + (0.309 + 0.951i)15-s + (5.31 − 3.86i)17-s + (−1.59 + 4.90i)19-s − 2.34·21-s + 4.96·23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−2.43 − 7.49i)29-s + (0.285 + 0.207i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (−0.361 + 0.262i)5-s + (−0.274 − 0.843i)7-s + (−0.269 − 0.195i)9-s + (0.179 + 0.983i)11-s + (0.888 + 0.645i)13-s + (0.0797 + 0.245i)15-s + (1.28 − 0.937i)17-s + (−0.365 + 1.12i)19-s − 0.512·21-s + 1.03·23-s + (0.0618 − 0.190i)25-s + (−0.155 + 0.113i)27-s + (−0.452 − 1.39i)29-s + (0.0511 + 0.0371i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.739 + 0.673i$
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: \((2,\ 1320,\ (\ :1/2),\ 0.739 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690126568\)
\(L(\frac12)\) \(\approx\) \(1.690126568\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.594 - 3.26i)T \)
good7 \( 1 + (0.725 + 2.23i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.20 - 2.32i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.31 + 3.86i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.59 - 4.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.96T + 23T^{2} \)
29 \( 1 + (2.43 + 7.49i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.285 - 0.207i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.213 - 0.656i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.23 + 9.96i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.22T + 43T^{2} \)
47 \( 1 + (-2.61 + 8.04i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.889 + 0.646i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.15 + 3.54i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.55 + 4.03i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.807T + 67T^{2} \)
71 \( 1 + (3.64 - 2.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.97 - 6.07i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.78 - 2.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.7 - 7.84i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (-5.76 - 4.19i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.612733061794564147383381648644, −8.661409350646068051612437971889, −7.66355340843463847408507372985, −7.22371182042078319361606985298, −6.45784179666889679703292238810, −5.42542505150582795752186492892, −4.12750544957671388512501175142, −3.53744902906747488725207819889, −2.17717877037335624435453258547, −0.893286164660894611004664090309, 1.10022533118022057220262029178, 2.92938685759280189438743076674, 3.44217540321913136743377185389, 4.62431376841692857463326263777, 5.64179600688159939314467355019, 6.13275205907146045061219387139, 7.43932684398966834469593981265, 8.412688323354380153629910509777, 8.828322076870724834059988877993, 9.534407007734985600576928807289

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