Properties

Label 2-1176-1.1-c3-0-49
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 + 13.3·5-s + 9·9-s − 1.42·11-s − 38.7·13-s − 40.0·15-s − 27.3·17-s + 65.4·19-s + 2.54·23-s + 52.9·25-s − 27·27-s − 63.7·29-s − 51.9·31-s + 4.27·33-s − 335.·37-s + 116.·39-s − 447.·41-s − 170.·43-s + 120.·45-s + 116.·47-s + 82.0·51-s + 86.3·53-s − 19.0·55-s − 196.·57-s − 380.·59-s + 199.·61-s − 517.·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.19·5-s + 0.333·9-s − 0.0390·11-s − 0.827·13-s − 0.688·15-s − 0.390·17-s + 0.790·19-s + 0.0230·23-s + 0.423·25-s − 0.192·27-s − 0.408·29-s − 0.301·31-s + 0.0225·33-s − 1.49·37-s + 0.477·39-s − 1.70·41-s − 0.604·43-s + 0.397·45-s + 0.362·47-s + 0.225·51-s + 0.223·53-s − 0.0466·55-s − 0.456·57-s − 0.840·59-s + 0.419·61-s − 0.986·65-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 \( 1 - 13.3T + 125T^{2} \)
11 \( 1 + 1.42T + 1.33e3T^{2} \)
13 \( 1 + 38.7T + 2.19e3T^{2} \)
17 \( 1 + 27.3T + 4.91e3T^{2} \)
19 \( 1 - 65.4T + 6.85e3T^{2} \)
23 \( 1 - 2.54T + 1.21e4T^{2} \)
29 \( 1 + 63.7T + 2.43e4T^{2} \)
31 \( 1 + 51.9T + 2.97e4T^{2} \)
37 \( 1 + 335.T + 5.06e4T^{2} \)
41 \( 1 + 447.T + 6.89e4T^{2} \)
43 \( 1 + 170.T + 7.95e4T^{2} \)
47 \( 1 - 116.T + 1.03e5T^{2} \)
53 \( 1 - 86.3T + 1.48e5T^{2} \)
59 \( 1 + 380.T + 2.05e5T^{2} \)
61 \( 1 - 199.T + 2.26e5T^{2} \)
67 \( 1 - 951.T + 3.00e5T^{2} \)
71 \( 1 - 830.T + 3.57e5T^{2} \)
73 \( 1 - 332.T + 3.89e5T^{2} \)
79 \( 1 - 755.T + 4.93e5T^{2} \)
83 \( 1 - 15.2T + 5.71e5T^{2} \)
89 \( 1 + 1.55e3T + 7.04e5T^{2} \)
97 \( 1 + 101.T + 9.12e5T^{2} \)
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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

−9.254555512004426082651524427805, −8.185827220745813221662645139032, −7.07311395084747386650666967213, −6.50675497314069263681047861932, −5.37233903081176296172297453510, −5.11029871405416478283511505132, −3.68690951252763864924660131651, −2.39456186353322551859158932999, −1.47462031569902553606617484185, 0, 1.47462031569902553606617484185, 2.39456186353322551859158932999, 3.68690951252763864924660131651, 5.11029871405416478283511505132, 5.37233903081176296172297453510, 6.50675497314069263681047861932, 7.07311395084747386650666967213, 8.185827220745813221662645139032, 9.254555512004426082651524427805

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