Properties

Label 2-1176-1.1-c1-0-18
Degree $2$
Conductor $1176$
Sign $-1$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 9-s − 6·11-s + 3·13-s − 2·15-s − 4·17-s + 5·19-s − 4·23-s − 25-s + 27-s − 4·29-s − 7·31-s − 6·33-s − 9·37-s + 3·39-s + 2·41-s − 43-s − 2·45-s − 2·47-s − 4·51-s + 8·53-s + 12·55-s + 5·57-s − 10·61-s − 6·65-s − 15·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.80·11-s + 0.832·13-s − 0.516·15-s − 0.970·17-s + 1.14·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.25·31-s − 1.04·33-s − 1.47·37-s + 0.480·39-s + 0.312·41-s − 0.152·43-s − 0.298·45-s − 0.291·47-s − 0.560·51-s + 1.09·53-s + 1.61·55-s + 0.662·57-s − 1.28·61-s − 0.744·65-s − 1.83·67-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: $-1$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.202774031586236845269371818770, −8.469023022179139867424912270992, −7.68752876422389994182638168187, −7.27717359877612471793627114412, −5.90935721239845183282820059742, −5.01036761283716733705466627986, −3.91987293239119779975337919921, −3.15658926737264229996933544486, −1.96192269343325068864778164904, 0, 1.96192269343325068864778164904, 3.15658926737264229996933544486, 3.91987293239119779975337919921, 5.01036761283716733705466627986, 5.90935721239845183282820059742, 7.27717359877612471793627114412, 7.68752876422389994182638168187, 8.469023022179139867424912270992, 9.202774031586236845269371818770

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