Properties

Label 2-48e2-1.1-c1-0-10
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  + 2.82·5-s − 2.82·7-s − 4·11-s + 5.65·13-s + 2·17-s + 4·19-s − 5.65·23-s + 3.00·25-s − 2.82·29-s + 8.48·31-s − 8.00·35-s + 10·41-s + 12·43-s + 5.65·47-s + 1.00·49-s − 2.82·53-s − 11.3·55-s + 4·59-s − 11.3·61-s + 16.0·65-s + 4·67-s + 5.65·71-s + 2·73-s + 11.3·77-s + 8.48·79-s + 4·83-s + 5.65·85-s + ⋯
L(s)  = 1  + 1.26·5-s − 1.06·7-s − 1.20·11-s + 1.56·13-s + 0.485·17-s + 0.917·19-s − 1.17·23-s + 0.600·25-s − 0.525·29-s + 1.52·31-s − 1.35·35-s + 1.56·41-s + 1.82·43-s + 0.825·47-s + 0.142·49-s − 0.388·53-s − 1.52·55-s + 0.520·59-s − 1.44·61-s + 1.98·65-s + 0.488·67-s + 0.671·71-s + 0.234·73-s + 1.28·77-s + 0.954·79-s + 0.439·83-s + 0.613·85-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.055975314\)
\(L(\frac12)\) \(\approx\) \(2.055975314\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + 2.82T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.327795282091750540174519367152, −8.202722803732038260819914072311, −7.54298636928153294021071163057, −6.31560238075764798982347559811, −5.99165327885955224849963000095, −5.34922354924852502163277944794, −4.05054195852994500497621289189, −3.07802410675241370947638387044, −2.27750845644887222463745971397, −0.947900956292784675730688449337, 0.947900956292784675730688449337, 2.27750845644887222463745971397, 3.07802410675241370947638387044, 4.05054195852994500497621289189, 5.34922354924852502163277944794, 5.99165327885955224849963000095, 6.31560238075764798982347559811, 7.54298636928153294021071163057, 8.202722803732038260819914072311, 9.327795282091750540174519367152

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