Properties

Degree 2
Conductor 5
Sign $1$
Motivic weight 33
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 9.61e4·2-s + 1.07e8·3-s + 6.61e8·4-s + 1.52e11·5-s − 1.03e13·6-s − 8.61e13·7-s + 7.62e14·8-s + 5.94e15·9-s − 1.46e16·10-s + 9.57e16·11-s + 7.09e16·12-s + 2.44e18·13-s + 8.28e18·14-s + 1.63e19·15-s − 7.90e19·16-s − 6.06e19·17-s − 5.72e20·18-s − 1.50e21·19-s + 1.00e20·20-s − 9.23e21·21-s − 9.20e21·22-s + 1.88e22·23-s + 8.18e22·24-s + 2.32e22·25-s − 2.35e23·26-s + 4.17e22·27-s − 5.69e22·28-s + ⋯
L(s)  = 1  − 1.03·2-s + 1.43·3-s + 0.0770·4-s + 0.447·5-s − 1.49·6-s − 0.979·7-s + 0.957·8-s + 1.07·9-s − 0.464·10-s + 0.628·11-s + 0.110·12-s + 1.01·13-s + 1.01·14-s + 0.643·15-s − 1.07·16-s − 0.302·17-s − 1.11·18-s − 1.19·19-s + 0.0344·20-s − 1.40·21-s − 0.651·22-s + 0.639·23-s + 1.37·24-s + 0.200·25-s − 1.05·26-s + 0.100·27-s − 0.0754·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(33\)
character  :  $\chi_{5} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :33/2),\ 1)$
$L(17)$  $\approx$  $1.878608457$
$L(\frac12)$  $\approx$  $1.878608457$
$L(\frac{35}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 5$,\(F_p(T)\) is a polynomial of degree 2. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 - 1.52e11T \)
good2 \( 1 + 9.61e4T + 8.58e9T^{2} \)
3 \( 1 - 1.07e8T + 5.55e15T^{2} \)
7 \( 1 + 8.61e13T + 7.73e27T^{2} \)
11 \( 1 - 9.57e16T + 2.32e34T^{2} \)
13 \( 1 - 2.44e18T + 5.75e36T^{2} \)
17 \( 1 + 6.06e19T + 4.02e40T^{2} \)
19 \( 1 + 1.50e21T + 1.58e42T^{2} \)
23 \( 1 - 1.88e22T + 8.65e44T^{2} \)
29 \( 1 - 2.67e24T + 1.81e48T^{2} \)
31 \( 1 - 2.64e24T + 1.64e49T^{2} \)
37 \( 1 + 1.12e26T + 5.63e51T^{2} \)
41 \( 1 - 7.31e26T + 1.66e53T^{2} \)
43 \( 1 - 3.88e26T + 8.02e53T^{2} \)
47 \( 1 - 3.28e27T + 1.51e55T^{2} \)
53 \( 1 - 3.35e28T + 7.96e56T^{2} \)
59 \( 1 - 2.93e29T + 2.74e58T^{2} \)
61 \( 1 - 7.07e28T + 8.23e58T^{2} \)
67 \( 1 - 1.99e30T + 1.82e60T^{2} \)
71 \( 1 + 1.15e30T + 1.23e61T^{2} \)
73 \( 1 + 2.83e30T + 3.08e61T^{2} \)
79 \( 1 - 6.93e30T + 4.18e62T^{2} \)
83 \( 1 - 4.09e31T + 2.13e63T^{2} \)
89 \( 1 + 1.49e32T + 2.13e64T^{2} \)
97 \( 1 - 1.00e33T + 3.65e65T^{2} \)
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

−15.88226551089039131829352915912, −14.13360754624482032271256871517, −13.10805041089935188953883518446, −10.35824314641015837234367958754, −9.124358829806207380489282908370, −8.466111735709032767421405763693, −6.68459592403554418143647784869, −3.95397766072631363283846176999, −2.44002294241966640068758474033, −0.956507833322988145561840779123, 0.956507833322988145561840779123, 2.44002294241966640068758474033, 3.95397766072631363283846176999, 6.68459592403554418143647784869, 8.466111735709032767421405763693, 9.124358829806207380489282908370, 10.35824314641015837234367958754, 13.10805041089935188953883518446, 14.13360754624482032271256871517, 15.88226551089039131829352915912

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