Properties

Label 2-2496-1.1-c3-0-71
Degree $2$
Conductor $2496$
Sign $-1$
Analytic cond. $147.268$
Root an. cond. $12.1354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3.29·5-s − 25.8·7-s + 9·9-s + 8.83·11-s + 13·13-s + 9.87·15-s − 10.2·17-s + 119.·19-s + 77.6·21-s − 141.·23-s − 114.·25-s − 27·27-s − 170.·29-s + 226.·31-s − 26.5·33-s + 85.1·35-s + 225.·37-s − 39·39-s − 274.·41-s + 111.·43-s − 29.6·45-s + 156.·47-s + 326.·49-s + 30.7·51-s + 85.1·53-s − 29.0·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.294·5-s − 1.39·7-s + 0.333·9-s + 0.242·11-s + 0.277·13-s + 0.169·15-s − 0.146·17-s + 1.44·19-s + 0.806·21-s − 1.28·23-s − 0.913·25-s − 0.192·27-s − 1.09·29-s + 1.31·31-s − 0.139·33-s + 0.411·35-s + 1.00·37-s − 0.160·39-s − 1.04·41-s + 0.394·43-s − 0.0981·45-s + 0.485·47-s + 0.951·49-s + 0.0844·51-s + 0.220·53-s − 0.0712·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(147.268\)
Root analytic conductor: \(12.1354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2496,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
13 \( 1 - 13T \)
good5 \( 1 + 3.29T + 125T^{2} \)
7 \( 1 + 25.8T + 343T^{2} \)
11 \( 1 - 8.83T + 1.33e3T^{2} \)
17 \( 1 + 10.2T + 4.91e3T^{2} \)
19 \( 1 - 119.T + 6.85e3T^{2} \)
23 \( 1 + 141.T + 1.21e4T^{2} \)
29 \( 1 + 170.T + 2.43e4T^{2} \)
31 \( 1 - 226.T + 2.97e4T^{2} \)
37 \( 1 - 225.T + 5.06e4T^{2} \)
41 \( 1 + 274.T + 6.89e4T^{2} \)
43 \( 1 - 111.T + 7.95e4T^{2} \)
47 \( 1 - 156.T + 1.03e5T^{2} \)
53 \( 1 - 85.1T + 1.48e5T^{2} \)
59 \( 1 - 889.T + 2.05e5T^{2} \)
61 \( 1 + 463.T + 2.26e5T^{2} \)
67 \( 1 - 459.T + 3.00e5T^{2} \)
71 \( 1 + 560.T + 3.57e5T^{2} \)
73 \( 1 - 784.T + 3.89e5T^{2} \)
79 \( 1 - 241.T + 4.93e5T^{2} \)
83 \( 1 - 1.27e3T + 5.71e5T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + 79.9T + 9.12e5T^{2} \)
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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

−8.045030971501394029901190526248, −7.38638032112470281840826131679, −6.47740486592304785958528969408, −6.01521157931770596600837732441, −5.16668359020135727828426824063, −3.98385001002060120631303679655, −3.48602073263167174314829164135, −2.32615848931917361610287461346, −0.950807194171427108330702342892, 0, 0.950807194171427108330702342892, 2.32615848931917361610287461346, 3.48602073263167174314829164135, 3.98385001002060120631303679655, 5.16668359020135727828426824063, 6.01521157931770596600837732441, 6.47740486592304785958528969408, 7.38638032112470281840826131679, 8.045030971501394029901190526248

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