Properties

Label 2-3072-1.1-c1-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.41·5-s + 9-s − 4·11-s + 1.41·13-s − 1.41·15-s − 4·19-s + 5.65·23-s − 2.99·25-s + 27-s + 7.07·29-s + 5.65·31-s − 4·33-s − 4.24·37-s + 1.41·39-s − 12·43-s − 1.41·45-s − 11.3·47-s − 7·49-s − 1.41·53-s + 5.65·55-s − 4·57-s + 4·59-s − 12.7·61-s − 2.00·65-s − 4·67-s + 5.65·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.632·5-s + 0.333·9-s − 1.20·11-s + 0.392·13-s − 0.365·15-s − 0.917·19-s + 1.17·23-s − 0.599·25-s + 0.192·27-s + 1.31·29-s + 1.01·31-s − 0.696·33-s − 0.697·37-s + 0.226·39-s − 1.82·43-s − 0.210·45-s − 1.65·47-s − 49-s − 0.194·53-s + 0.762·55-s − 0.529·57-s + 0.520·59-s − 1.62·61-s − 0.248·65-s − 0.488·67-s + 0.681·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 1.41T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.352722555401997626130774544649, −7.77619444184807694434240628838, −6.91188857190002773971050090137, −6.19637127914246934061891987990, −5.00362272253802268899164325088, −4.49160535687503510785015772604, −3.35403216284626023545799053666, −2.79164589099848667553436036959, −1.57076231192331961601946483294, 0, 1.57076231192331961601946483294, 2.79164589099848667553436036959, 3.35403216284626023545799053666, 4.49160535687503510785015772604, 5.00362272253802268899164325088, 6.19637127914246934061891987990, 6.91188857190002773971050090137, 7.77619444184807694434240628838, 8.352722555401997626130774544649

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