Properties

Label 2-2496-1.1-c3-0-85
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 − 2.77·5-s − 5.23·7-s + 9·9-s − 26.0·11-s + 13·13-s + 8.32·15-s + 4.06·17-s + 73.9·19-s + 15.6·21-s − 145.·23-s − 117.·25-s − 27·27-s + 259.·29-s + 78.5·31-s + 78.0·33-s + 14.5·35-s − 99.8·37-s − 39·39-s + 346.·41-s + 137.·43-s − 24.9·45-s + 56.7·47-s − 315.·49-s − 12.1·51-s − 173.·53-s + 72.2·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.248·5-s − 0.282·7-s + 0.333·9-s − 0.713·11-s + 0.277·13-s + 0.143·15-s + 0.0579·17-s + 0.893·19-s + 0.163·21-s − 1.31·23-s − 0.938·25-s − 0.192·27-s + 1.66·29-s + 0.455·31-s + 0.411·33-s + 0.0701·35-s − 0.443·37-s − 0.160·39-s + 1.31·41-s + 0.488·43-s − 0.0827·45-s + 0.176·47-s − 0.920·49-s − 0.0334·51-s − 0.449·53-s + 0.177·55-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 \( 1 + 2.77T + 125T^{2} \)
7 \( 1 + 5.23T + 343T^{2} \)
11 \( 1 + 26.0T + 1.33e3T^{2} \)
17 \( 1 - 4.06T + 4.91e3T^{2} \)
19 \( 1 - 73.9T + 6.85e3T^{2} \)
23 \( 1 + 145.T + 1.21e4T^{2} \)
29 \( 1 - 259.T + 2.43e4T^{2} \)
31 \( 1 - 78.5T + 2.97e4T^{2} \)
37 \( 1 + 99.8T + 5.06e4T^{2} \)
41 \( 1 - 346.T + 6.89e4T^{2} \)
43 \( 1 - 137.T + 7.95e4T^{2} \)
47 \( 1 - 56.7T + 1.03e5T^{2} \)
53 \( 1 + 173.T + 1.48e5T^{2} \)
59 \( 1 + 675.T + 2.05e5T^{2} \)
61 \( 1 - 798.T + 2.26e5T^{2} \)
67 \( 1 - 292.T + 3.00e5T^{2} \)
71 \( 1 + 333.T + 3.57e5T^{2} \)
73 \( 1 - 243.T + 3.89e5T^{2} \)
79 \( 1 + 670.T + 4.93e5T^{2} \)
83 \( 1 - 786.T + 5.71e5T^{2} \)
89 \( 1 + 317.T + 7.04e5T^{2} \)
97 \( 1 - 886.T + 9.12e5T^{2} \)
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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

−7.982478270124590159127077990747, −7.57234548875484384333073274301, −6.48154264950208503416041401362, −5.93348875997616272664582114323, −5.07026924190639187302453605560, −4.24931374690436510704310812883, −3.31962697441669604665039528626, −2.29463016680673451048576056544, −1.02408165677062171441291786225, 0, 1.02408165677062171441291786225, 2.29463016680673451048576056544, 3.31962697441669604665039528626, 4.24931374690436510704310812883, 5.07026924190639187302453605560, 5.93348875997616272664582114323, 6.48154264950208503416041401362, 7.57234548875484384333073274301, 7.982478270124590159127077990747

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