Properties

Degree 2
Conductor 2
Sign $1$
Motivic weight 49
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.67e7·2-s + 7.94e11·3-s + 2.81e14·4-s − 5.65e16·5-s + 1.33e19·6-s − 4.38e20·7-s + 4.72e21·8-s + 3.91e23·9-s − 9.48e23·10-s + 5.68e25·11-s + 2.23e26·12-s + 1.61e27·13-s − 7.35e27·14-s − 4.49e28·15-s + 7.92e28·16-s + 2.39e30·17-s + 6.57e30·18-s − 9.22e30·19-s − 1.59e31·20-s − 3.48e32·21-s + 9.53e32·22-s − 1.45e33·23-s + 3.75e33·24-s − 1.45e34·25-s + 2.71e34·26-s + 1.21e35·27-s − 1.23e35·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.62·3-s + 0.5·4-s − 0.424·5-s + 1.14·6-s − 0.865·7-s + 0.353·8-s + 1.63·9-s − 0.299·10-s + 1.73·11-s + 0.811·12-s + 0.827·13-s − 0.611·14-s − 0.688·15-s + 0.250·16-s + 1.71·17-s + 1.15·18-s − 0.432·19-s − 0.212·20-s − 1.40·21-s + 1.22·22-s − 0.633·23-s + 0.574·24-s − 0.820·25-s + 0.584·26-s + 1.03·27-s − 0.432·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(49\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :49/2),\ 1)$
$L(25)$  $\approx$  $5.337800343$
$L(\frac12)$  $\approx$  $5.337800343$
$L(\frac{51}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - 1.67e7T \)
good3 \( 1 - 7.94e11T + 2.39e23T^{2} \)
5 \( 1 + 5.65e16T + 1.77e34T^{2} \)
7 \( 1 + 4.38e20T + 2.56e41T^{2} \)
11 \( 1 - 5.68e25T + 1.06e51T^{2} \)
13 \( 1 - 1.61e27T + 3.83e54T^{2} \)
17 \( 1 - 2.39e30T + 1.95e60T^{2} \)
19 \( 1 + 9.22e30T + 4.55e62T^{2} \)
23 \( 1 + 1.45e33T + 5.30e66T^{2} \)
29 \( 1 - 1.26e36T + 4.54e71T^{2} \)
31 \( 1 + 2.70e36T + 1.19e73T^{2} \)
37 \( 1 + 7.30e37T + 6.94e76T^{2} \)
41 \( 1 - 1.21e39T + 1.06e79T^{2} \)
43 \( 1 + 1.71e40T + 1.09e80T^{2} \)
47 \( 1 + 1.04e41T + 8.56e81T^{2} \)
53 \( 1 + 1.17e41T + 3.08e84T^{2} \)
59 \( 1 + 8.71e42T + 5.91e86T^{2} \)
61 \( 1 + 2.61e43T + 3.02e87T^{2} \)
67 \( 1 - 3.87e44T + 3.00e89T^{2} \)
71 \( 1 - 2.06e44T + 5.14e90T^{2} \)
73 \( 1 - 4.21e44T + 2.00e91T^{2} \)
79 \( 1 - 8.65e45T + 9.63e92T^{2} \)
83 \( 1 - 1.57e47T + 1.08e94T^{2} \)
89 \( 1 + 2.35e47T + 3.31e95T^{2} \)
97 \( 1 - 1.57e48T + 2.24e97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.20456799217905896884768467306, −14.73896957976704895223707120740, −13.75322535842257672037216602098, −12.16424367654486949592547232511, −9.687572339335342823233636254144, −8.188336359086953354755005804921, −6.51203288617538966076572647104, −3.87923887090072069028369432781, −3.25043929444362311762323158344, −1.48282513455735403202035086127, 1.48282513455735403202035086127, 3.25043929444362311762323158344, 3.87923887090072069028369432781, 6.51203288617538966076572647104, 8.188336359086953354755005804921, 9.687572339335342823233636254144, 12.16424367654486949592547232511, 13.75322535842257672037216602098, 14.73896957976704895223707120740, 16.20456799217905896884768467306

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