Properties

Label 2-2496-1.1-c3-0-57
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4.51·5-s − 7.48·7-s + 9·9-s + 66.8·11-s + 13·13-s − 13.5·15-s + 96.9·17-s − 31.4·19-s − 22.4·21-s + 183.·23-s − 104.·25-s + 27·27-s − 112.·29-s − 77.2·31-s + 200.·33-s + 33.7·35-s − 54.7·37-s + 39·39-s + 451.·41-s + 113.·43-s − 40.6·45-s − 42.2·47-s − 287·49-s + 290.·51-s + 530.·53-s − 302.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.403·5-s − 0.404·7-s + 0.333·9-s + 1.83·11-s + 0.277·13-s − 0.233·15-s + 1.38·17-s − 0.380·19-s − 0.233·21-s + 1.66·23-s − 0.836·25-s + 0.192·27-s − 0.718·29-s − 0.447·31-s + 1.05·33-s + 0.163·35-s − 0.243·37-s + 0.160·39-s + 1.72·41-s + 0.402·43-s − 0.134·45-s − 0.131·47-s − 0.836·49-s + 0.798·51-s + 1.37·53-s − 0.740·55-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: \(0\)
Selberg data: \((2,\ 2496,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.174666197\)
\(L(\frac12)\) \(\approx\) \(3.174666197\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.51T + 125T^{2} \)
7 \( 1 + 7.48T + 343T^{2} \)
11 \( 1 - 66.8T + 1.33e3T^{2} \)
17 \( 1 - 96.9T + 4.91e3T^{2} \)
19 \( 1 + 31.4T + 6.85e3T^{2} \)
23 \( 1 - 183.T + 1.21e4T^{2} \)
29 \( 1 + 112.T + 2.43e4T^{2} \)
31 \( 1 + 77.2T + 2.97e4T^{2} \)
37 \( 1 + 54.7T + 5.06e4T^{2} \)
41 \( 1 - 451.T + 6.89e4T^{2} \)
43 \( 1 - 113.T + 7.95e4T^{2} \)
47 \( 1 + 42.2T + 1.03e5T^{2} \)
53 \( 1 - 530.T + 1.48e5T^{2} \)
59 \( 1 + 219.T + 2.05e5T^{2} \)
61 \( 1 + 822.T + 2.26e5T^{2} \)
67 \( 1 - 872.T + 3.00e5T^{2} \)
71 \( 1 + 100.T + 3.57e5T^{2} \)
73 \( 1 + 165.T + 3.89e5T^{2} \)
79 \( 1 + 545.T + 4.93e5T^{2} \)
83 \( 1 - 454.T + 5.71e5T^{2} \)
89 \( 1 + 230.T + 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−8.712446002861329549094184021003, −7.74973996056545857948559214668, −7.15030542240181342270984373072, −6.34840367405229737082237076615, −5.53700433353291577516224393089, −4.30000680635409416599517501628, −3.70143962593896823233969191797, −3.00884997751659826751595371950, −1.67029514541797575544766709986, −0.812221806099329517431611034269, 0.812221806099329517431611034269, 1.67029514541797575544766709986, 3.00884997751659826751595371950, 3.70143962593896823233969191797, 4.30000680635409416599517501628, 5.53700433353291577516224393089, 6.34840367405229737082237076615, 7.15030542240181342270984373072, 7.74973996056545857948559214668, 8.712446002861329549094184021003

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