Properties

Label 2-1152-4.3-c2-0-2
Degree $2$
Conductor $1152$
Sign $-1$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.46·5-s + 5.48i·7-s + 10.5i·11-s − 8.96·13-s − 16.2·17-s − 2.96i·19-s − 1.46i·23-s − 18.9·25-s − 25.0·29-s − 10.5i·31-s + 13.5i·35-s + 16.9·37-s − 29.0·41-s − 34.9i·43-s − 86.1i·47-s + ⋯
L(s)  = 1  + 0.492·5-s + 0.783i·7-s + 0.962i·11-s − 0.689·13-s − 0.955·17-s − 0.156i·19-s − 0.0635i·23-s − 0.757·25-s − 0.865·29-s − 0.339i·31-s + 0.385i·35-s + 0.458·37-s − 0.707·41-s − 0.813i·43-s − 1.83i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4118362283\)
\(L(\frac12)\) \(\approx\) \(0.4118362283\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 - 2.46T + 25T^{2} \)
7 \( 1 - 5.48iT - 49T^{2} \)
11 \( 1 - 10.5iT - 121T^{2} \)
13 \( 1 + 8.96T + 169T^{2} \)
17 \( 1 + 16.2T + 289T^{2} \)
19 \( 1 + 2.96iT - 361T^{2} \)
23 \( 1 + 1.46iT - 529T^{2} \)
29 \( 1 + 25.0T + 841T^{2} \)
31 \( 1 + 10.5iT - 961T^{2} \)
37 \( 1 - 16.9T + 1.36e3T^{2} \)
41 \( 1 + 29.0T + 1.68e3T^{2} \)
43 \( 1 + 34.9iT - 1.84e3T^{2} \)
47 \( 1 + 86.1iT - 2.20e3T^{2} \)
53 \( 1 + 80.1T + 2.80e3T^{2} \)
59 \( 1 - 66.4iT - 3.48e3T^{2} \)
61 \( 1 - 0.966T + 3.72e3T^{2} \)
67 \( 1 - 113. iT - 4.48e3T^{2} \)
71 \( 1 + 90.5iT - 5.04e3T^{2} \)
73 \( 1 - 51.7T + 5.32e3T^{2} \)
79 \( 1 + 80.4iT - 6.24e3T^{2} \)
83 \( 1 - 79.9iT - 6.88e3T^{2} \)
89 \( 1 + 142.T + 7.92e3T^{2} \)
97 \( 1 + 45.8T + 9.40e3T^{2} \)
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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

−9.838330036724251656216045932442, −9.316811392132444480184167070541, −8.499056295446328326349997691194, −7.46065661353247103174716280677, −6.71376749270037751812764372462, −5.72926763117069217894187563188, −4.98085069343103275275268611014, −3.97760644373346888982195941584, −2.51487126690261679437617563828, −1.88438877099315770561840317276, 0.11295561804468839637723482487, 1.56219875144283458474411231081, 2.81393917996652912960625962013, 3.90909901955442397979229453990, 4.85903610039301446697602974396, 5.88597483677872065078760960965, 6.62910100496553485310176512791, 7.57682414699611012794495313833, 8.321600655618771920154026780985, 9.384420348914755511148742616139

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