Properties

Label 2-3072-1.1-c1-0-1
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 − 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: \(0\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4411116808\)
\(L(\frac12)\) \(\approx\) \(0.4411116808\)
\(L(\frac{3}{2})\) 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

−8.691570209726364465816336509650, −7.951796059363429919239668160432, −7.00638938704548957779032535845, −6.38764093318650211638703888602, −5.87907232179393807089773081520, −4.83147318091589642998273263721, −4.02043268620955719180630533188, −3.07501098781709785572695988197, −2.19585836955146979442182775257, −0.38558982328734312295465831128, 0.38558982328734312295465831128, 2.19585836955146979442182775257, 3.07501098781709785572695988197, 4.02043268620955719180630533188, 4.83147318091589642998273263721, 5.87907232179393807089773081520, 6.38764093318650211638703888602, 7.00638938704548957779032535845, 7.951796059363429919239668160432, 8.691570209726364465816336509650

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