Properties

Label 2-1176-1.1-c3-0-32
Degree $2$
Conductor $1176$
Sign $-1$
Analytic cond. $69.3862$
Root an. cond. $8.32984$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 14.6·5-s + 9·9-s − 11.4·11-s − 20.8·13-s + 43.9·15-s + 18.8·17-s + 91.0·19-s + 28.9·23-s + 89.3·25-s − 27·27-s + 281.·29-s − 276.·31-s + 34.3·33-s + 250.·37-s + 62.4·39-s + 12.1·41-s + 65.2·43-s − 131.·45-s + 199.·47-s − 56.4·51-s − 64.1·53-s + 167.·55-s − 273.·57-s − 69.0·59-s − 656.·61-s + 304.·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.30·5-s + 0.333·9-s − 0.313·11-s − 0.443·13-s + 0.756·15-s + 0.268·17-s + 1.09·19-s + 0.262·23-s + 0.715·25-s − 0.192·27-s + 1.80·29-s − 1.59·31-s + 0.181·33-s + 1.11·37-s + 0.256·39-s + 0.0462·41-s + 0.231·43-s − 0.436·45-s + 0.620·47-s − 0.155·51-s − 0.166·53-s + 0.410·55-s − 0.634·57-s − 0.152·59-s − 1.37·61-s + 0.581·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(69.3862\)
Root analytic conductor: \(8.32984\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1176,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
7 \( 1 \)
good5 \( 1 + 14.6T + 125T^{2} \)
11 \( 1 + 11.4T + 1.33e3T^{2} \)
13 \( 1 + 20.8T + 2.19e3T^{2} \)
17 \( 1 - 18.8T + 4.91e3T^{2} \)
19 \( 1 - 91.0T + 6.85e3T^{2} \)
23 \( 1 - 28.9T + 1.21e4T^{2} \)
29 \( 1 - 281.T + 2.43e4T^{2} \)
31 \( 1 + 276.T + 2.97e4T^{2} \)
37 \( 1 - 250.T + 5.06e4T^{2} \)
41 \( 1 - 12.1T + 6.89e4T^{2} \)
43 \( 1 - 65.2T + 7.95e4T^{2} \)
47 \( 1 - 199.T + 1.03e5T^{2} \)
53 \( 1 + 64.1T + 1.48e5T^{2} \)
59 \( 1 + 69.0T + 2.05e5T^{2} \)
61 \( 1 + 656.T + 2.26e5T^{2} \)
67 \( 1 + 419.T + 3.00e5T^{2} \)
71 \( 1 - 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + 103.T + 3.89e5T^{2} \)
79 \( 1 - 260.T + 4.93e5T^{2} \)
83 \( 1 - 879.T + 5.71e5T^{2} \)
89 \( 1 + 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + 177.T + 9.12e5T^{2} \)
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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

−8.989063324800842866741456612012, −7.87196858728890511488429327873, −7.51153785292212178178202186486, −6.56908464541230493977556745870, −5.46110886442174881266709207314, −4.68086414501444989275640126367, −3.77379123344185623429667452278, −2.76445278587140312267906967610, −1.07318458867128794535154598641, 0, 1.07318458867128794535154598641, 2.76445278587140312267906967610, 3.77379123344185623429667452278, 4.68086414501444989275640126367, 5.46110886442174881266709207314, 6.56908464541230493977556745870, 7.51153785292212178178202186486, 7.87196858728890511488429327873, 8.989063324800842866741456612012

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