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  − 1.38e15·2-s + 4.94e24·3-s − 8.21e30·4-s − 1.47e36·5-s − 6.86e39·6-s + 3.27e43·7-s + 2.54e46·8-s + 1.05e49·9-s + 2.04e51·10-s − 1.87e52·11-s − 4.06e55·12-s − 1.80e57·13-s − 4.54e58·14-s − 7.30e60·15-s + 4.80e61·16-s + 4.46e61·17-s − 1.46e64·18-s − 1.38e66·19-s + 1.21e67·20-s + 1.62e68·21-s + 2.59e67·22-s + 1.24e70·23-s + 1.25e71·24-s + 1.19e72·25-s + 2.50e72·26-s − 1.65e73·27-s − 2.69e74·28-s + ⋯
L(s)  = 1  − 0.435·2-s + 1.32·3-s − 0.810·4-s − 1.48·5-s − 0.577·6-s + 0.984·7-s + 0.788·8-s + 0.759·9-s + 0.647·10-s − 0.0436·11-s − 1.07·12-s − 0.775·13-s − 0.428·14-s − 1.97·15-s + 0.467·16-s + 0.0191·17-s − 0.330·18-s − 1.93·19-s + 1.20·20-s + 1.30·21-s + 0.0190·22-s + 0.922·23-s + 1.04·24-s + 1.20·25-s + 0.337·26-s − 0.318·27-s − 0.797·28-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  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $1.440286859$
$L(\frac12)$  $\approx$  $1.440286859$
$L(\frac{105}{2})$   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 \( 1 + 1.38e15T + 1.01e31T^{2} \)
3 \( 1 - 4.94e24T + 1.39e49T^{2} \)
5 \( 1 + 1.47e36T + 9.86e71T^{2} \)
7 \( 1 - 3.27e43T + 1.10e87T^{2} \)
11 \( 1 + 1.87e52T + 1.83e107T^{2} \)
13 \( 1 + 1.80e57T + 5.44e114T^{2} \)
17 \( 1 - 4.46e61T + 5.44e126T^{2} \)
19 \( 1 + 1.38e66T + 5.14e131T^{2} \)
23 \( 1 - 1.24e70T + 1.81e140T^{2} \)
29 \( 1 + 3.27e75T + 4.23e150T^{2} \)
31 \( 1 - 9.03e76T + 4.07e153T^{2} \)
37 \( 1 - 3.03e80T + 3.34e161T^{2} \)
41 \( 1 - 1.50e83T + 1.30e166T^{2} \)
43 \( 1 - 2.40e84T + 1.76e168T^{2} \)
47 \( 1 + 3.60e84T + 1.68e172T^{2} \)
53 \( 1 + 9.21e87T + 3.98e177T^{2} \)
59 \( 1 - 1.63e91T + 2.49e182T^{2} \)
61 \( 1 + 9.20e91T + 7.74e183T^{2} \)
67 \( 1 - 1.05e94T + 1.21e188T^{2} \)
71 \( 1 - 3.92e95T + 4.78e190T^{2} \)
73 \( 1 - 2.83e95T + 8.36e191T^{2} \)
79 \( 1 - 2.16e97T + 2.85e195T^{2} \)
83 \( 1 - 4.78e98T + 4.62e197T^{2} \)
89 \( 1 - 3.22e100T + 6.12e200T^{2} \)
97 \( 1 - 7.81e101T + 4.33e204T^{2} \)
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

−14.42102432161500767847016529900, −12.78880608308057498000357628597, −11.01006999598865038448858092830, −9.172162614807167093763513029501, −8.179495383170566436088472123815, −7.61150408980178002530619878066, −4.63298857090053863915474771330, −3.84454976406587141845199562543, −2.28341398809701646391944965828, −0.63196233004117782406842765128, 0.63196233004117782406842765128, 2.28341398809701646391944965828, 3.84454976406587141845199562543, 4.63298857090053863915474771330, 7.61150408980178002530619878066, 8.179495383170566436088472123815, 9.172162614807167093763513029501, 11.01006999598865038448858092830, 12.78880608308057498000357628597, 14.42102432161500767847016529900

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