Properties

Label 2-9576-1.1-c1-0-47
Degree $2$
Conductor $9576$
Sign $1$
Analytic cond. $76.4647$
Root an. cond. $8.74441$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.841·5-s + 7-s + 4.29·11-s + 5.02·13-s − 6.86·17-s + 19-s + 3.02·23-s − 4.29·25-s + 2.18·29-s + 7.31·31-s − 0.841·35-s + 3.27·37-s + 0.317·41-s + 12.7·47-s + 49-s − 6.18·53-s − 3.61·55-s − 14.1·59-s + 6·61-s − 4.22·65-s + 1.86·67-s + 7.11·71-s − 2.74·73-s + 4.29·77-s + 1.72·79-s + 6.72·83-s + 5.77·85-s + ⋯
L(s)  = 1  − 0.376·5-s + 0.377·7-s + 1.29·11-s + 1.39·13-s − 1.66·17-s + 0.229·19-s + 0.630·23-s − 0.858·25-s + 0.404·29-s + 1.31·31-s − 0.142·35-s + 0.537·37-s + 0.0495·41-s + 1.85·47-s + 0.142·49-s − 0.849·53-s − 0.487·55-s − 1.84·59-s + 0.768·61-s − 0.524·65-s + 0.227·67-s + 0.843·71-s − 0.321·73-s + 0.489·77-s + 0.194·79-s + 0.737·83-s + 0.626·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9576\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(76.4647\)
Root analytic conductor: \(8.74441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.398038730\)
\(L(\frac12)\) \(\approx\) \(2.398038730\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + 0.841T + 5T^{2} \)
11 \( 1 - 4.29T + 11T^{2} \)
13 \( 1 - 5.02T + 13T^{2} \)
17 \( 1 + 6.86T + 17T^{2} \)
23 \( 1 - 3.02T + 23T^{2} \)
29 \( 1 - 2.18T + 29T^{2} \)
31 \( 1 - 7.31T + 31T^{2} \)
37 \( 1 - 3.27T + 37T^{2} \)
41 \( 1 - 0.317T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 + 6.18T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 1.86T + 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + 2.74T + 73T^{2} \)
79 \( 1 - 1.72T + 79T^{2} \)
83 \( 1 - 6.72T + 83T^{2} \)
89 \( 1 + 3.29T + 89T^{2} \)
97 \( 1 + 1.43T + 97T^{2} \)
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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

−7.78758249649793040473142073992, −6.84334460090149493467086134094, −6.41788214412229552699449733916, −5.80747903310820188104155475867, −4.70078021297811194839134753248, −4.20281252449408683236140833305, −3.60599521298646817845975881874, −2.60536847620827166720106523175, −1.60610180456851709967054703843, −0.789532545732488832147249260937, 0.789532545732488832147249260937, 1.60610180456851709967054703843, 2.60536847620827166720106523175, 3.60599521298646817845975881874, 4.20281252449408683236140833305, 4.70078021297811194839134753248, 5.80747903310820188104155475867, 6.41788214412229552699449733916, 6.84334460090149493467086134094, 7.78758249649793040473142073992

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