Properties

Label 2-1176-24.5-c0-0-2
Degree $2$
Conductor $1176$
Sign $1$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 11-s − 12-s − 15-s + 16-s − 18-s + 20-s − 22-s + 24-s − 27-s + 29-s + 30-s − 31-s − 32-s − 33-s + 36-s − 40-s + 44-s + 45-s − 48-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 11-s − 12-s − 15-s + 16-s − 18-s + 20-s − 22-s + 24-s − 27-s + 29-s + 30-s − 31-s − 32-s − 33-s + 36-s − 40-s + 44-s + 45-s − 48-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1176} (197, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6425046856\)
\(L(\frac12)\) \(\approx\) \(0.6425046856\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 - T + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−9.912736823186251069305112354368, −9.387382770402993747373823412182, −8.518603449405621620195511856016, −7.35100983172926698446391031517, −6.61081794820260449350823538573, −6.01754372167287612008607827805, −5.18647715230684529380301050857, −3.80239448679299552640242337277, −2.21974967604645260546398307986, −1.18516225414688112075824373034, 1.18516225414688112075824373034, 2.21974967604645260546398307986, 3.80239448679299552640242337277, 5.18647715230684529380301050857, 6.01754372167287612008607827805, 6.61081794820260449350823538573, 7.35100983172926698446391031517, 8.518603449405621620195511856016, 9.387382770402993747373823412182, 9.912736823186251069305112354368

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