Properties

Label 2-2496-1.1-c1-0-42
Degree $2$
Conductor $2496$
Sign $-1$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
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 + 3.23·5-s − 0.763·7-s + 9-s − 4.47·11-s + 13-s − 3.23·15-s − 4.47·17-s + 3.23·19-s + 0.763·21-s − 6.47·23-s + 5.47·25-s − 27-s − 0.472·29-s − 4.76·31-s + 4.47·33-s − 2.47·35-s − 4.47·37-s − 39-s − 4.76·41-s − 2.47·43-s + 3.23·45-s − 8.47·47-s − 6.41·49-s + 4.47·51-s + 8.47·53-s − 14.4·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.44·5-s − 0.288·7-s + 0.333·9-s − 1.34·11-s + 0.277·13-s − 0.835·15-s − 1.08·17-s + 0.742·19-s + 0.166·21-s − 1.34·23-s + 1.09·25-s − 0.192·27-s − 0.0876·29-s − 0.855·31-s + 0.778·33-s − 0.417·35-s − 0.735·37-s − 0.160·39-s − 0.744·41-s − 0.376·43-s + 0.482·45-s − 1.23·47-s − 0.916·49-s + 0.626·51-s + 1.16·53-s − 1.95·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 + 0.763T + 7T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 3.23T + 19T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + 0.472T + 29T^{2} \)
31 \( 1 + 4.76T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 4.76T + 41T^{2} \)
43 \( 1 + 2.47T + 43T^{2} \)
47 \( 1 + 8.47T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 5.70T + 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 2.29T + 89T^{2} \)
97 \( 1 - 7.52T + 97T^{2} \)
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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

−8.622743141324375129590504447764, −7.73514599627840100681766414843, −6.76977355717461548567420947736, −6.15588997456836058887207522205, −5.41540144659690839147801261852, −4.92264205118945584159750549089, −3.61349696638074206546179088610, −2.44512008565931364788640421093, −1.68975572598307039033093284927, 0, 1.68975572598307039033093284927, 2.44512008565931364788640421093, 3.61349696638074206546179088610, 4.92264205118945584159750549089, 5.41540144659690839147801261852, 6.15588997456836058887207522205, 6.76977355717461548567420947736, 7.73514599627840100681766414843, 8.622743141324375129590504447764

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