Properties

Label 2-3960-1320.299-c0-0-4
Degree $2$
Conductor $3960$
Sign $0.700 + 0.713i$
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  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.183 + 0.253i)7-s + (0.809 − 0.587i)8-s + 0.999i·10-s + (−0.987 + 0.156i)11-s + (−1.69 − 0.550i)13-s + (−0.183 − 0.253i)14-s + (0.309 + 0.951i)16-s + (−0.363 − 0.5i)19-s + (−0.951 − 0.309i)20-s + (0.156 − 0.987i)22-s − 1.61i·23-s + (0.809 − 0.587i)25-s + (1.04 − 1.44i)26-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.183 + 0.253i)7-s + (0.809 − 0.587i)8-s + 0.999i·10-s + (−0.987 + 0.156i)11-s + (−1.69 − 0.550i)13-s + (−0.183 − 0.253i)14-s + (0.309 + 0.951i)16-s + (−0.363 − 0.5i)19-s + (−0.951 − 0.309i)20-s + (0.156 − 0.987i)22-s − 1.61i·23-s + (0.809 − 0.587i)25-s + (1.04 − 1.44i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.700 + 0.713i$
Analytic conductor: \(1.97629\)
Root analytic conductor: \(1.40580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (1619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :0),\ 0.700 + 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7246886684\)
\(L(\frac12)\) \(\approx\) \(0.7246886684\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 \)
5 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.987 - 0.156i)T \)
good7 \( 1 + (0.183 - 0.253i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (1.69 + 0.550i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.61iT - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.59 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + 1.97iT - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−8.482574747722734675760319290621, −7.82117781790747488862686464160, −7.05719413122585773083628233118, −6.36662764978008248958202451359, −5.58948555045842812317350395169, −4.96964983808520353156904825269, −4.46412963115344089580883952733, −2.80091412680833316876344820724, −2.12352777116065178785801289378, −0.43695981798257153839927964775, 1.43214921403448145097296640256, 2.42499591466296938701961123239, 2.90754120762772124436827162102, 4.04989958509836348092191642003, 4.94217013518638586312485494492, 5.56082314087060207625491007027, 6.56182604501615659225934368215, 7.62069204225928416710377310953, 7.83356033690376033295804298731, 9.128175038588413475063138372960

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