Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 103
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 6.07e15·2-s + 6.42e24·3-s + 2.67e31·4-s − 2.83e35·5-s + 3.90e40·6-s − 2.16e43·7-s + 1.00e47·8-s + 2.73e49·9-s − 1.72e51·10-s + 2.34e53·11-s + 1.71e56·12-s − 1.57e57·13-s − 1.31e59·14-s − 1.82e60·15-s + 3.40e62·16-s − 1.46e63·17-s + 1.66e65·18-s + 1.14e66·19-s − 7.58e66·20-s − 1.38e68·21-s + 1.42e69·22-s + 3.42e69·23-s + 6.47e71·24-s − 9.05e71·25-s − 9.53e72·26-s + 8.63e73·27-s − 5.78e74·28-s + ⋯
L(s)  = 1  + 1.90·2-s + 1.72·3-s + 2.63·4-s − 0.285·5-s + 3.28·6-s − 0.649·7-s + 3.11·8-s + 1.96·9-s − 0.545·10-s + 0.548·11-s + 4.53·12-s − 0.673·13-s − 1.23·14-s − 0.492·15-s + 3.31·16-s − 0.625·17-s + 3.74·18-s + 1.59·19-s − 0.753·20-s − 1.11·21-s + 1.04·22-s + 0.254·23-s + 5.37·24-s − 0.918·25-s − 1.28·26-s + 1.66·27-s − 1.71·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(104-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+51.5) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(103\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $12.87119385$
$L(\frac12)$  $\approx$  $12.87119385$
$L(\frac{105}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 - 6.07e15T + 1.01e31T^{2} \)
3 \( 1 - 6.42e24T + 1.39e49T^{2} \)
5 \( 1 + 2.83e35T + 9.86e71T^{2} \)
7 \( 1 + 2.16e43T + 1.10e87T^{2} \)
11 \( 1 - 2.34e53T + 1.83e107T^{2} \)
13 \( 1 + 1.57e57T + 5.44e114T^{2} \)
17 \( 1 + 1.46e63T + 5.44e126T^{2} \)
19 \( 1 - 1.14e66T + 5.14e131T^{2} \)
23 \( 1 - 3.42e69T + 1.81e140T^{2} \)
29 \( 1 - 3.20e74T + 4.23e150T^{2} \)
31 \( 1 - 5.53e76T + 4.07e153T^{2} \)
37 \( 1 + 8.68e79T + 3.34e161T^{2} \)
41 \( 1 - 6.58e82T + 1.30e166T^{2} \)
43 \( 1 + 2.31e84T + 1.76e168T^{2} \)
47 \( 1 + 2.37e86T + 1.68e172T^{2} \)
53 \( 1 + 1.71e88T + 3.98e177T^{2} \)
59 \( 1 + 1.30e91T + 2.49e182T^{2} \)
61 \( 1 + 6.15e91T + 7.74e183T^{2} \)
67 \( 1 - 6.14e92T + 1.21e188T^{2} \)
71 \( 1 - 1.58e95T + 4.78e190T^{2} \)
73 \( 1 + 1.01e96T + 8.36e191T^{2} \)
79 \( 1 - 8.88e97T + 2.85e195T^{2} \)
83 \( 1 + 4.44e98T + 4.62e197T^{2} \)
89 \( 1 - 2.52e100T + 6.12e200T^{2} \)
97 \( 1 - 6.67e101T + 4.33e204T^{2} \)
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

−14.00759299947461153652013899083, −13.17309733043186448538303677324, −11.83198775069337076932663323564, −9.696149810581833566809529906624, −7.75555231094391419035026214450, −6.61812074826167301001942548511, −4.73471342165889258499251803665, −3.53520829839401559070020806391, −2.92680892065378442959232696076, −1.72487038180223718843796543554, 1.72487038180223718843796543554, 2.92680892065378442959232696076, 3.53520829839401559070020806391, 4.73471342165889258499251803665, 6.61812074826167301001942548511, 7.75555231094391419035026214450, 9.696149810581833566809529906624, 11.83198775069337076932663323564, 13.17309733043186448538303677324, 14.00759299947461153652013899083

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