Properties

Label 2-3072-1.1-c1-0-38
Degree $2$
Conductor $3072$
Sign $1$
Analytic cond. $24.5300$
Root an. cond. $4.95278$
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  + 3-s + 2.82·5-s + 4.24·7-s + 9-s − 4·11-s + 4.24·13-s + 2.82·15-s + 6·17-s − 2·19-s + 4.24·21-s + 2.82·23-s + 3.00·25-s + 27-s − 5.65·29-s − 4.24·31-s − 4·33-s + 12·35-s + 4.24·37-s + 4.24·39-s − 10·41-s + 6·43-s + 2.82·45-s − 2.82·47-s + 10.9·49-s + 6·51-s − 5.65·53-s − 11.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.26·5-s + 1.60·7-s + 0.333·9-s − 1.20·11-s + 1.17·13-s + 0.730·15-s + 1.45·17-s − 0.458·19-s + 0.925·21-s + 0.589·23-s + 0.600·25-s + 0.192·27-s − 1.05·29-s − 0.762·31-s − 0.696·33-s + 2.02·35-s + 0.697·37-s + 0.679·39-s − 1.56·41-s + 0.914·43-s + 0.421·45-s − 0.412·47-s + 1.57·49-s + 0.840·51-s − 0.777·53-s − 1.52·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $1$
Analytic conductor: \(24.5300\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.680577813\)
\(L(\frac12)\) \(\approx\) \(3.680577813\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 + 4.24T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 4.24T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−8.635159546709504234667338907807, −7.974570990862250186745901337450, −7.51674537177975832348418944940, −6.32189295784899135292004637526, −5.42787690158571511922408982857, −5.14823138510250543370890819612, −3.95435075669513884682492463610, −2.91306687234717475144541813677, −1.92648141866716423382181490437, −1.33647034865909251669042359181, 1.33647034865909251669042359181, 1.92648141866716423382181490437, 2.91306687234717475144541813677, 3.95435075669513884682492463610, 5.14823138510250543370890819612, 5.42787690158571511922408982857, 6.32189295784899135292004637526, 7.51674537177975832348418944940, 7.974570990862250186745901337450, 8.635159546709504234667338907807

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