Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} \cdot 17^{2} $
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  − 4·7-s − 3·9-s − 2·11-s + 6·13-s + 6·29-s − 6·31-s − 2·37-s + 6·41-s − 6·43-s + 10·47-s + 9·49-s + 6·53-s − 10·61-s + 12·63-s + 2·67-s − 6·71-s + 6·73-s + 8·77-s − 6·79-s + 9·81-s − 6·83-s − 18·89-s − 24·91-s − 14·97-s + 6·99-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.51·7-s − 9-s − 0.603·11-s + 1.66·13-s + 1.11·29-s − 1.07·31-s − 0.328·37-s + 0.937·41-s − 0.914·43-s + 1.45·47-s + 9/7·49-s + 0.824·53-s − 1.28·61-s + 1.51·63-s + 0.244·67-s − 0.712·71-s + 0.702·73-s + 0.911·77-s − 0.675·79-s + 81-s − 0.658·83-s − 1.90·89-s − 2.51·91-s − 1.42·97-s + 0.603·99-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28900\)    =    \(2^{2} \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{28900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 28900,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.031486854$
$L(\frac12)$  $\approx$  $1.031486854$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−15.43586136189884, −14.58916151971381, −14.01507452607379, −13.49129175555503, −13.20960628158896, −12.51391557135563, −12.11822002080806, −11.34666275268929, −10.82621763142558, −10.43863357081689, −9.752936326236378, −9.129722920170606, −8.671526821268346, −8.235749621077719, −7.394875219058654, −6.781764397512214, −6.143569894285307, −5.816696422288335, −5.226417159850517, −4.160623152918800, −3.634831043342376, −2.974557506213160, −2.553734888146299, −1.371177014618828, −0.4020151321399599, 0.4020151321399599, 1.371177014618828, 2.553734888146299, 2.974557506213160, 3.634831043342376, 4.160623152918800, 5.226417159850517, 5.816696422288335, 6.143569894285307, 6.781764397512214, 7.394875219058654, 8.235749621077719, 8.671526821268346, 9.129722920170606, 9.752936326236378, 10.43863357081689, 10.82621763142558, 11.34666275268929, 12.11822002080806, 12.51391557135563, 13.20960628158896, 13.49129175555503, 14.01507452607379, 14.58916151971381, 15.43586136189884

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