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Properties

Label 2-1176-21.20-c1-0-37
Degree $2$
Conductor $1176$
Sign $-0.557 + 0.830i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
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  + (−0.454 − 1.67i)3-s + 2.80·5-s + (−2.58 + 1.52i)9-s − 5.48i·11-s + 1.35i·13-s + (−1.27 − 4.69i)15-s − 5.77·17-s − 1.98i·19-s − 2.42i·23-s + 2.88·25-s + (3.71 + 3.63i)27-s − 7.05i·29-s − 3.55i·31-s + (−9.15 + 2.49i)33-s + 4.28·37-s + ⋯
L(s)  = 1  + (−0.262 − 0.964i)3-s + 1.25·5-s + (−0.862 + 0.506i)9-s − 1.65i·11-s + 0.376i·13-s + (−0.329 − 1.21i)15-s − 1.40·17-s − 0.455i·19-s − 0.505i·23-s + 0.576·25-s + (0.715 + 0.698i)27-s − 1.31i·29-s − 0.637i·31-s + (−1.59 + 0.434i)33-s + 0.704·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.557 + 0.830i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.497521016\)
\(L(\frac12)\) \(\approx\) \(1.497521016\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.454 + 1.67i)T \)
7 \( 1 \)
good5 \( 1 - 2.80T + 5T^{2} \)
11 \( 1 + 5.48iT - 11T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 + 5.77T + 17T^{2} \)
19 \( 1 + 1.98iT - 19T^{2} \)
23 \( 1 + 2.42iT - 23T^{2} \)
29 \( 1 + 7.05iT - 29T^{2} \)
31 \( 1 + 3.55iT - 31T^{2} \)
37 \( 1 - 4.28T + 37T^{2} \)
41 \( 1 + 1.81T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 - 0.402T + 47T^{2} \)
53 \( 1 + 6.09iT - 53T^{2} \)
59 \( 1 + 2.56T + 59T^{2} \)
61 \( 1 - 5.49iT - 61T^{2} \)
67 \( 1 + 6.90T + 67T^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 + 0.341iT - 73T^{2} \)
79 \( 1 + 2.38T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 - 1.15T + 89T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−9.285581347943890789147403444467, −8.740834765859591800666460663804, −7.86456816540509510457680428502, −6.74911500312096041052663506222, −6.10077413041581532125981645555, −5.66450007651442380559340323199, −4.37847571032433299789590534476, −2.77961881562257226282823211360, −2.04972852231126135230841229114, −0.64168489230661189716147005802, 1.74948122039068341836620277695, 2.80514245005041912319210355508, 4.18459341954939892407756126031, 4.91898825299466932196222171745, 5.73269284545815544498052223396, 6.55105473557913562184463030039, 7.49348385747420411768184464393, 8.865406134679280323661238383048, 9.317871929757971393253468805399, 10.09140771585883900485031638468

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