Properties

Label 2-1176-1.1-c3-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 16.6·5-s + 9·9-s − 49.2·11-s − 64.4·13-s − 49.8·15-s − 132.·17-s + 82.0·19-s + 82.0·23-s + 150.·25-s − 27·27-s + 157.·29-s + 185.·31-s + 147.·33-s − 51.9·37-s + 193.·39-s + 49.4·41-s + 313.·43-s + 149.·45-s + 553.·47-s + 397.·51-s − 619.·53-s − 817.·55-s − 246.·57-s + 712.·59-s + 287.·61-s − 1.07e3·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.48·5-s + 0.333·9-s − 1.34·11-s − 1.37·13-s − 0.857·15-s − 1.88·17-s + 0.990·19-s + 0.743·23-s + 1.20·25-s − 0.192·27-s + 1.00·29-s + 1.07·31-s + 0.779·33-s − 0.230·37-s + 0.794·39-s + 0.188·41-s + 1.11·43-s + 0.494·45-s + 1.71·47-s + 1.09·51-s − 1.60·53-s − 2.00·55-s − 0.572·57-s + 1.57·59-s + 0.603·61-s − 2.04·65-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: \(0\)
Selberg data: \((2,\ 1176,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.805563699\)
\(L(\frac12)\) \(\approx\) \(1.805563699\)
\(L(\frac{5}{2})\) 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 \( 1 - 16.6T + 125T^{2} \)
11 \( 1 + 49.2T + 1.33e3T^{2} \)
13 \( 1 + 64.4T + 2.19e3T^{2} \)
17 \( 1 + 132.T + 4.91e3T^{2} \)
19 \( 1 - 82.0T + 6.85e3T^{2} \)
23 \( 1 - 82.0T + 1.21e4T^{2} \)
29 \( 1 - 157.T + 2.43e4T^{2} \)
31 \( 1 - 185.T + 2.97e4T^{2} \)
37 \( 1 + 51.9T + 5.06e4T^{2} \)
41 \( 1 - 49.4T + 6.89e4T^{2} \)
43 \( 1 - 313.T + 7.95e4T^{2} \)
47 \( 1 - 553.T + 1.03e5T^{2} \)
53 \( 1 + 619.T + 1.48e5T^{2} \)
59 \( 1 - 712.T + 2.05e5T^{2} \)
61 \( 1 - 287.T + 2.26e5T^{2} \)
67 \( 1 - 226.T + 3.00e5T^{2} \)
71 \( 1 - 55.3T + 3.57e5T^{2} \)
73 \( 1 + 799.T + 3.89e5T^{2} \)
79 \( 1 + 120.T + 4.93e5T^{2} \)
83 \( 1 - 857.T + 5.71e5T^{2} \)
89 \( 1 - 377.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−9.557807595395788735533749920912, −8.780855562956937417448733857940, −7.56736704596782613166475585575, −6.80622178673015292518999943172, −5.96436667969269227134938034229, −5.11869965497779920503891244566, −4.64629378547434903352906280493, −2.71570203186089850101471631962, −2.20651834931504747098513911918, −0.69044925793705892987760497981, 0.69044925793705892987760497981, 2.20651834931504747098513911918, 2.71570203186089850101471631962, 4.64629378547434903352906280493, 5.11869965497779920503891244566, 5.96436667969269227134938034229, 6.80622178673015292518999943172, 7.56736704596782613166475585575, 8.780855562956937417448733857940, 9.557807595395788735533749920912

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