Properties

Label 2-1176-1.1-c3-0-33
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 − 14.5·5-s + 9·9-s + 10.1·11-s + 0.623·13-s + 43.5·15-s + 16.8·17-s − 75.7·19-s + 46.7·23-s + 85.5·25-s − 27·27-s + 73.5·29-s + 135.·31-s − 30.3·33-s + 133.·37-s − 1.87·39-s + 69.3·41-s + 279.·43-s − 130.·45-s − 372.·47-s − 50.4·51-s + 656.·53-s − 146.·55-s + 227.·57-s + 730.·59-s − 39.5·61-s − 9.05·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.29·5-s + 0.333·9-s + 0.277·11-s + 0.0133·13-s + 0.749·15-s + 0.239·17-s − 0.915·19-s + 0.423·23-s + 0.684·25-s − 0.192·27-s + 0.470·29-s + 0.784·31-s − 0.160·33-s + 0.591·37-s − 0.00768·39-s + 0.263·41-s + 0.990·43-s − 0.432·45-s − 1.15·47-s − 0.138·51-s + 1.70·53-s − 0.359·55-s + 0.528·57-s + 1.61·59-s − 0.0829·61-s − 0.0172·65-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 \( 1 + 14.5T + 125T^{2} \)
11 \( 1 - 10.1T + 1.33e3T^{2} \)
13 \( 1 - 0.623T + 2.19e3T^{2} \)
17 \( 1 - 16.8T + 4.91e3T^{2} \)
19 \( 1 + 75.7T + 6.85e3T^{2} \)
23 \( 1 - 46.7T + 1.21e4T^{2} \)
29 \( 1 - 73.5T + 2.43e4T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 - 133.T + 5.06e4T^{2} \)
41 \( 1 - 69.3T + 6.89e4T^{2} \)
43 \( 1 - 279.T + 7.95e4T^{2} \)
47 \( 1 + 372.T + 1.03e5T^{2} \)
53 \( 1 - 656.T + 1.48e5T^{2} \)
59 \( 1 - 730.T + 2.05e5T^{2} \)
61 \( 1 + 39.5T + 2.26e5T^{2} \)
67 \( 1 - 417.T + 3.00e5T^{2} \)
71 \( 1 + 831.T + 3.57e5T^{2} \)
73 \( 1 + 1.19e3T + 3.89e5T^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 117.T + 5.71e5T^{2} \)
89 \( 1 - 678.T + 7.04e5T^{2} \)
97 \( 1 + 1.65e3T + 9.12e5T^{2} \)
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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

−8.828442746174199415525385839245, −8.147531351919606811817165556910, −7.30072673562718736942980585146, −6.57400593537917437694237385984, −5.58755090346612643593921234088, −4.47146115771470077234563565350, −3.93956394909069476051115920787, −2.70561639700481847680246692637, −1.09364473732739488342367524078, 0, 1.09364473732739488342367524078, 2.70561639700481847680246692637, 3.93956394909069476051115920787, 4.47146115771470077234563565350, 5.58755090346612643593921234088, 6.57400593537917437694237385984, 7.30072673562718736942980585146, 8.147531351919606811817165556910, 8.828442746174199415525385839245

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