Properties

Label 2-3072-1.1-c1-0-55
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 + 0.473·5-s − 4.55·7-s + 9-s + 3.49·11-s + 0.0840·13-s + 0.473·15-s − 3.61·17-s + 3.61·19-s − 4.55·21-s − 2.82·23-s − 4.77·25-s + 27-s − 7.30·29-s + 0.557·31-s + 3.49·33-s − 2.15·35-s − 6.20·37-s + 0.0840·39-s − 9.27·41-s + 2.27·43-s + 0.473·45-s + 2.82·47-s + 13.7·49-s − 3.61·51-s − 0.697·53-s + 1.65·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.211·5-s − 1.72·7-s + 0.333·9-s + 1.05·11-s + 0.0233·13-s + 0.122·15-s − 0.877·17-s + 0.829·19-s − 0.994·21-s − 0.589·23-s − 0.955·25-s + 0.192·27-s − 1.35·29-s + 0.100·31-s + 0.608·33-s − 0.364·35-s − 1.01·37-s + 0.0134·39-s − 1.44·41-s + 0.347·43-s + 0.0706·45-s + 0.412·47-s + 1.96·49-s − 0.506·51-s − 0.0958·53-s + 0.223·55-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: \(1\)
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 \( 1 - 0.473T + 5T^{2} \)
7 \( 1 + 4.55T + 7T^{2} \)
11 \( 1 - 3.49T + 11T^{2} \)
13 \( 1 - 0.0840T + 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 - 3.61T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 7.30T + 29T^{2} \)
31 \( 1 - 0.557T + 31T^{2} \)
37 \( 1 + 6.20T + 37T^{2} \)
41 \( 1 + 9.27T + 41T^{2} \)
43 \( 1 - 2.27T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 0.697T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 - 3.85T + 61T^{2} \)
67 \( 1 + 5.33T + 67T^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 + 0.541T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−8.582028680850034841995780654397, −7.41640972469585181714386535841, −6.84549398301460030937788896567, −6.19544695089386236649720187164, −5.40787075207221840632782540766, −4.00913488076193079190011117964, −3.63582143723551907606364880325, −2.67788671682274415243091658748, −1.62553312189158187587009641796, 0, 1.62553312189158187587009641796, 2.67788671682274415243091658748, 3.63582143723551907606364880325, 4.00913488076193079190011117964, 5.40787075207221840632782540766, 6.19544695089386236649720187164, 6.84549398301460030937788896567, 7.41640972469585181714386535841, 8.582028680850034841995780654397

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