Properties

Label 2-2496-1.1-c1-0-11
Degree $2$
Conductor $2496$
Sign $1$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·5-s − 4·7-s + 9-s + 2·11-s + 13-s − 4·15-s − 6·17-s − 4·19-s + 4·21-s + 4·23-s + 11·25-s − 27-s + 6·29-s + 8·31-s − 2·33-s − 16·35-s + 10·37-s − 39-s − 4·41-s + 4·43-s + 4·45-s − 6·47-s + 9·49-s + 6·51-s − 6·53-s + 8·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 1.03·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.348·33-s − 2.70·35-s + 1.64·37-s − 0.160·39-s − 0.624·41-s + 0.609·43-s + 0.596·45-s − 0.875·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 1.07·55-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771788425\)
\(L(\frac12)\) \(\approx\) \(1.771788425\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.209632797526139354099334044597, −8.412081168783159735694253984596, −6.79976403646101930660690744681, −6.38643960892890947570913822009, −6.24285194557670647480016843307, −5.09754766043068360145456374630, −4.25781876644228347131524734341, −2.95526170418117470667875421111, −2.18289937290554630887181547294, −0.876666400119120651748065595063, 0.876666400119120651748065595063, 2.18289937290554630887181547294, 2.95526170418117470667875421111, 4.25781876644228347131524734341, 5.09754766043068360145456374630, 6.24285194557670647480016843307, 6.38643960892890947570913822009, 6.79976403646101930660690744681, 8.412081168783159735694253984596, 9.209632797526139354099334044597

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