Properties

Label 2-1176-1.1-c3-0-47
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 + 7.81·5-s + 9·9-s + 42.7·11-s + 10.9·13-s − 23.4·15-s − 80.2·17-s − 112.·19-s − 70.1·23-s − 63.9·25-s − 27·27-s − 147.·29-s + 144.·31-s − 128.·33-s − 0.292·37-s − 32.8·39-s + 294.·41-s + 337.·43-s + 70.3·45-s − 104.·47-s + 240.·51-s − 143.·53-s + 334.·55-s + 338.·57-s − 520.·59-s − 399.·61-s + 85.5·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.698·5-s + 0.333·9-s + 1.17·11-s + 0.233·13-s − 0.403·15-s − 1.14·17-s − 1.36·19-s − 0.636·23-s − 0.511·25-s − 0.192·27-s − 0.941·29-s + 0.837·31-s − 0.676·33-s − 0.00130·37-s − 0.134·39-s + 1.12·41-s + 1.19·43-s + 0.232·45-s − 0.324·47-s + 0.661·51-s − 0.370·53-s + 0.819·55-s + 0.785·57-s − 1.14·59-s − 0.839·61-s + 0.163·65-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 \( 1 - 7.81T + 125T^{2} \)
11 \( 1 - 42.7T + 1.33e3T^{2} \)
13 \( 1 - 10.9T + 2.19e3T^{2} \)
17 \( 1 + 80.2T + 4.91e3T^{2} \)
19 \( 1 + 112.T + 6.85e3T^{2} \)
23 \( 1 + 70.1T + 1.21e4T^{2} \)
29 \( 1 + 147.T + 2.43e4T^{2} \)
31 \( 1 - 144.T + 2.97e4T^{2} \)
37 \( 1 + 0.292T + 5.06e4T^{2} \)
41 \( 1 - 294.T + 6.89e4T^{2} \)
43 \( 1 - 337.T + 7.95e4T^{2} \)
47 \( 1 + 104.T + 1.03e5T^{2} \)
53 \( 1 + 143.T + 1.48e5T^{2} \)
59 \( 1 + 520.T + 2.05e5T^{2} \)
61 \( 1 + 399.T + 2.26e5T^{2} \)
67 \( 1 - 137.T + 3.00e5T^{2} \)
71 \( 1 - 266.T + 3.57e5T^{2} \)
73 \( 1 + 524.T + 3.89e5T^{2} \)
79 \( 1 - 433.T + 4.93e5T^{2} \)
83 \( 1 + 664.T + 5.71e5T^{2} \)
89 \( 1 + 803.T + 7.04e5T^{2} \)
97 \( 1 + 445.T + 9.12e5T^{2} \)
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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

−9.164715223784901298609552968392, −8.263657762248000292632601384206, −7.11856212956214989448365278186, −6.22197646837641412912503636679, −5.94017583673857056892758776546, −4.55555332543288288000929668643, −3.95275373496728053460901064445, −2.36183336160174760825939835424, −1.44329890736239276802597013967, 0, 1.44329890736239276802597013967, 2.36183336160174760825939835424, 3.95275373496728053460901064445, 4.55555332543288288000929668643, 5.94017583673857056892758776546, 6.22197646837641412912503636679, 7.11856212956214989448365278186, 8.263657762248000292632601384206, 9.164715223784901298609552968392

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