Properties

Degree 2
Conductor $ 5^{2} \cdot 241 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.49·2-s − 1.22·3-s + 4.20·4-s + 3.04·6-s − 0.136·7-s − 5.48·8-s − 1.50·9-s − 0.905·11-s − 5.13·12-s + 0.123·13-s + 0.339·14-s + 5.26·16-s − 1.26·17-s + 3.75·18-s − 2.13·19-s + 0.166·21-s + 2.25·22-s − 6.64·23-s + 6.70·24-s − 0.308·26-s + 5.50·27-s − 0.572·28-s + 5.36·29-s − 9.78·31-s − 2.13·32-s + 1.10·33-s + 3.13·34-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.705·3-s + 2.10·4-s + 1.24·6-s − 0.0514·7-s − 1.94·8-s − 0.502·9-s − 0.272·11-s − 1.48·12-s + 0.0343·13-s + 0.0906·14-s + 1.31·16-s − 0.305·17-s + 0.884·18-s − 0.489·19-s + 0.0363·21-s + 0.480·22-s − 1.38·23-s + 1.36·24-s − 0.0604·26-s + 1.05·27-s − 0.108·28-s + 0.996·29-s − 1.75·31-s − 0.377·32-s + 0.192·33-s + 0.538·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6025\)    =    \(5^{2} \cdot 241\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1859205452$
$L(\frac12)$  $\approx$  $0.1859205452$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;241\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;241\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 2.49T + 2T^{2} \)
3 \( 1 + 1.22T + 3T^{2} \)
7 \( 1 + 0.136T + 7T^{2} \)
11 \( 1 + 0.905T + 11T^{2} \)
13 \( 1 - 0.123T + 13T^{2} \)
17 \( 1 + 1.26T + 17T^{2} \)
19 \( 1 + 2.13T + 19T^{2} \)
23 \( 1 + 6.64T + 23T^{2} \)
29 \( 1 - 5.36T + 29T^{2} \)
31 \( 1 + 9.78T + 31T^{2} \)
37 \( 1 + 5.76T + 37T^{2} \)
41 \( 1 - 6.43T + 41T^{2} \)
43 \( 1 - 3.18T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 3.90T + 53T^{2} \)
59 \( 1 - 8.15T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 - 4.89T + 67T^{2} \)
71 \( 1 - 4.32T + 71T^{2} \)
73 \( 1 + 5.64T + 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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

−8.187498945855526615591571262592, −7.59469657741235213549529837197, −6.71616509761784313671102938160, −6.25491631843808871599599746739, −5.51670478869478182179342198943, −4.56366833790394423537593237656, −3.35327772551603931393493047998, −2.36662257398082114027121702059, −1.56648190417741679576757448773, −0.30394124144063704065703729827, 0.30394124144063704065703729827, 1.56648190417741679576757448773, 2.36662257398082114027121702059, 3.35327772551603931393493047998, 4.56366833790394423537593237656, 5.51670478869478182179342198943, 6.25491631843808871599599746739, 6.71616509761784313671102938160, 7.59469657741235213549529837197, 8.187498945855526615591571262592

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