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Properties

Label 2-1176-168.131-c0-0-7
Degree $2$
Conductor $1176$
Sign $-0.884 - 0.467i$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
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  + (−0.866 − 0.5i)2-s + (−0.793 − 0.608i)3-s + (0.499 + 0.866i)4-s + (0.382 + 0.923i)6-s − 0.999i·8-s + (0.258 + 0.965i)9-s + (−1.22 + 0.707i)11-s + (0.130 − 0.991i)12-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + (0.258 − 0.965i)18-s + (−1.60 − 0.923i)19-s + 1.41·22-s + (−0.608 + 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.793 − 0.608i)3-s + (0.499 + 0.866i)4-s + (0.382 + 0.923i)6-s − 0.999i·8-s + (0.258 + 0.965i)9-s + (−1.22 + 0.707i)11-s + (0.130 − 0.991i)12-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + (0.258 − 0.965i)18-s + (−1.60 − 0.923i)19-s + 1.41·22-s + (−0.608 + 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.884 - 0.467i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.884 - 0.467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08930371999\)
\(L(\frac12)\) \(\approx\) \(0.08930371999\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 0.765T + T^{2} \)
89 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.84iT - T^{2} \)
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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

−9.721094566611551355128046959677, −8.534522718022158933713314645625, −7.980596118322168430203276745117, −6.98561129897208764154771481728, −6.55522063209493133952082198347, −5.20804761636370410936685685389, −4.34031405729721114872381435747, −2.67627585321913736092580877048, −1.91972769300176594372404394106, −0.10666780879037050194736317975, 1.85032997645161176753826366508, 3.44902536001235715985422625768, 4.73788943289734943297899930221, 5.60809808122424168619837851700, 6.21441173957317362686339644197, 7.08120711401012467136624499348, 8.271683126550127075094431651737, 8.613373106689591981023793073360, 9.860805250366517948453844771178, 10.29736642110352884961565550511

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