Properties

Label 2-48e2-1.1-c3-0-6
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $135.940$
Root an. cond. $11.6593$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s − 12·7-s − 12·11-s − 20·13-s − 62·17-s − 108·19-s − 72·23-s − 61·25-s − 128·29-s − 204·31-s + 96·35-s + 228·37-s − 22·41-s + 204·43-s + 600·47-s − 199·49-s + 256·53-s + 96·55-s − 828·59-s + 84·61-s + 160·65-s − 348·67-s + 456·71-s − 822·73-s + 144·77-s − 1.35e3·79-s + 108·83-s + ⋯
L(s)  = 1  − 0.715·5-s − 0.647·7-s − 0.328·11-s − 0.426·13-s − 0.884·17-s − 1.30·19-s − 0.652·23-s − 0.487·25-s − 0.819·29-s − 1.18·31-s + 0.463·35-s + 1.01·37-s − 0.0838·41-s + 0.723·43-s + 1.86·47-s − 0.580·49-s + 0.663·53-s + 0.235·55-s − 1.82·59-s + 0.176·61-s + 0.305·65-s − 0.634·67-s + 0.762·71-s − 1.31·73-s + 0.213·77-s − 1.93·79-s + 0.142·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(135.940\)
Root analytic conductor: \(11.6593\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3838337575\)
\(L(\frac12)\) \(\approx\) \(0.3838337575\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + 8 T + p^{3} T^{2} \)
7 \( 1 + 12 T + p^{3} T^{2} \)
11 \( 1 + 12 T + p^{3} T^{2} \)
13 \( 1 + 20 T + p^{3} T^{2} \)
17 \( 1 + 62 T + p^{3} T^{2} \)
19 \( 1 + 108 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 + 128 T + p^{3} T^{2} \)
31 \( 1 + 204 T + p^{3} T^{2} \)
37 \( 1 - 228 T + p^{3} T^{2} \)
41 \( 1 + 22 T + p^{3} T^{2} \)
43 \( 1 - 204 T + p^{3} T^{2} \)
47 \( 1 - 600 T + p^{3} T^{2} \)
53 \( 1 - 256 T + p^{3} T^{2} \)
59 \( 1 + 828 T + p^{3} T^{2} \)
61 \( 1 - 84 T + p^{3} T^{2} \)
67 \( 1 + 348 T + p^{3} T^{2} \)
71 \( 1 - 456 T + p^{3} T^{2} \)
73 \( 1 + 822 T + p^{3} T^{2} \)
79 \( 1 + 1356 T + p^{3} T^{2} \)
83 \( 1 - 108 T + p^{3} T^{2} \)
89 \( 1 + 938 T + p^{3} T^{2} \)
97 \( 1 - 1278 T + p^{3} T^{2} \)
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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

−8.718408995599604414573737118981, −7.75586521190613843331410129779, −7.26077030590145632281536970552, −6.29769075781164828347322931537, −5.63452604508086372509024598996, −4.38030129514313824450107174268, −3.96053971539419950105277675155, −2.80860804162934035952720846965, −1.92241557506735767940215421638, −0.26006025946990651634413702473, 0.26006025946990651634413702473, 1.92241557506735767940215421638, 2.80860804162934035952720846965, 3.96053971539419950105277675155, 4.38030129514313824450107174268, 5.63452604508086372509024598996, 6.29769075781164828347322931537, 7.26077030590145632281536970552, 7.75586521190613843331410129779, 8.718408995599604414573737118981

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