Properties

Degree $2$
Conductor $1176$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.41·5-s + 9-s + 0.828·11-s − 4.24·13-s − 3.41·15-s + 7.41·17-s + 6.82·19-s − 4.82·23-s + 6.65·25-s − 27-s + 2.82·29-s − 2.82·31-s − 0.828·33-s − 1.65·37-s + 4.24·39-s + 10.2·41-s − 11.3·43-s + 3.41·45-s − 4.48·47-s − 7.41·51-s − 2·53-s + 2.82·55-s − 6.82·57-s + 8.48·59-s + 11.0·61-s − 14.4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.52·5-s + 0.333·9-s + 0.249·11-s − 1.17·13-s − 0.881·15-s + 1.79·17-s + 1.56·19-s − 1.00·23-s + 1.33·25-s − 0.192·27-s + 0.525·29-s − 0.508·31-s − 0.144·33-s − 0.272·37-s + 0.679·39-s + 1.59·41-s − 1.72·43-s + 0.508·45-s − 0.654·47-s − 1.03·51-s − 0.274·53-s + 0.381·55-s − 0.904·57-s + 1.10·59-s + 1.41·61-s − 1.79·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.860030798\)
\(L(\frac12)\) \(\approx\) \(1.860030798\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 3.41T + 5T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 7.41T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + 1.65T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 7.75T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 5.75T + 89T^{2} \)
97 \( 1 + 0.242T + 97T^{2} \)
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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

−9.957739283219519194481216297863, −9.355554026761523516885561775349, −8.022680413609285129565670462565, −7.19636794737525610329331931747, −6.27285204175584493663561643222, −5.44350299069127547691562540748, −5.06627991215715855586940941927, −3.50767919790606919660421938774, −2.29629771145379592239840068789, −1.14577477740938300051393507152, 1.14577477740938300051393507152, 2.29629771145379592239840068789, 3.50767919790606919660421938774, 5.06627991215715855586940941927, 5.44350299069127547691562540748, 6.27285204175584493663561643222, 7.19636794737525610329331931747, 8.022680413609285129565670462565, 9.355554026761523516885561775349, 9.957739283219519194481216297863

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