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Properties

Label 2-3960-1320.1019-c0-0-6
Degree $2$
Conductor $3960$
Sign $-0.100 - 0.994i$
Analytic cond. $1.97629$
Root an. cond. $1.40580$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.863 − 0.280i)7-s + (−0.309 + 0.951i)8-s + i·10-s + (0.891 + 0.453i)11-s + (0.183 − 0.253i)13-s + (0.863 + 0.280i)14-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)19-s + (−0.587 + 0.809i)20-s + (0.453 + 0.891i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (0.297 − 0.0966i)26-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.863 − 0.280i)7-s + (−0.309 + 0.951i)8-s + i·10-s + (0.891 + 0.453i)11-s + (0.183 − 0.253i)13-s + (0.863 + 0.280i)14-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)19-s + (−0.587 + 0.809i)20-s + (0.453 + 0.891i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (0.297 − 0.0966i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(1.97629\)
Root analytic conductor: \(1.40580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (2339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :0),\ -0.100 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.549947174\)
\(L(\frac12)\) \(\approx\) \(2.549947174\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 \)
5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.891 - 0.453i)T \)
good7 \( 1 + (-0.863 + 0.280i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.183 + 0.253i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 0.618iT - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.87 - 0.610i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - 1.78iT - T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−8.807491915289552295751793732338, −7.80423286775865219379145352274, −7.20234975922461442892803099946, −6.63827944296305866774618333215, −5.86833861869070588207112553280, −5.21486978240397544556660254663, −4.19599327187273566205726400798, −3.74038719501035370597211107948, −2.49039967657479490079090363752, −1.81178189391380518146111949468, 1.23222330980376701035337475041, 1.86257115170188356903929411334, 2.86937168367930113431151286147, 4.10856430617119427960983217967, 4.53631344044277202169194485122, 5.29776675753219828136171680578, 6.26481422649787733583005571691, 6.41570047444532325059317283578, 7.83514323841735608122372581779, 8.719694840354338502979960181854

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