Properties

Label 2-2496-1.1-c1-0-14
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·5-s + 2·7-s + 9-s − 4·11-s − 13-s − 4·15-s + 2·17-s − 2·19-s − 2·21-s + 11·25-s − 27-s + 6·29-s + 10·31-s + 4·33-s + 8·35-s − 10·37-s + 39-s + 8·41-s + 4·43-s + 4·45-s + 4·47-s − 3·49-s − 2·51-s + 10·53-s − 16·55-s + 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.03·15-s + 0.485·17-s − 0.458·19-s − 0.436·21-s + 11/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.696·33-s + 1.35·35-s − 1.64·37-s + 0.160·39-s + 1.24·41-s + 0.609·43-s + 0.596·45-s + 0.583·47-s − 3/7·49-s − 0.280·51-s + 1.37·53-s − 2.15·55-s + 0.264·57-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: $\chi_{2496} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.237549955\)
\(L(\frac12)\) \(\approx\) \(2.237549955\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.939639567543323946113513841442, −8.227808649057000200498276335540, −7.29595882657599730453498777162, −6.41649036742480464627329082855, −5.72491179107937457974413403419, −5.14896024739962723325634761707, −4.50008355692744503094190412349, −2.81997084527132682127236815913, −2.12656331686074115056549242086, −1.02919547182981641476270794586, 1.02919547182981641476270794586, 2.12656331686074115056549242086, 2.81997084527132682127236815913, 4.50008355692744503094190412349, 5.14896024739962723325634761707, 5.72491179107937457974413403419, 6.41649036742480464627329082855, 7.29595882657599730453498777162, 8.227808649057000200498276335540, 8.939639567543323946113513841442

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