Properties

Degree 2
Conductor $ 5^{2} \cdot 5077 $
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  + 2·2-s + 3·3-s + 2·4-s + 6·6-s + 4·7-s + 6·9-s − 6·11-s + 6·12-s + 4·13-s + 8·14-s − 4·16-s + 4·17-s + 12·18-s − 7·19-s + 12·21-s − 12·22-s + 6·23-s + 8·26-s + 9·27-s + 8·28-s − 6·29-s − 2·31-s − 8·32-s − 18·33-s + 8·34-s + 12·36-s − 14·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 4-s + 2.44·6-s + 1.51·7-s + 2·9-s − 1.80·11-s + 1.73·12-s + 1.10·13-s + 2.13·14-s − 16-s + 0.970·17-s + 2.82·18-s − 1.60·19-s + 2.61·21-s − 2.55·22-s + 1.25·23-s + 1.56·26-s + 1.73·27-s + 1.51·28-s − 1.11·29-s − 0.359·31-s − 1.41·32-s − 3.13·33-s + 1.37·34-s + 2·36-s − 2.27·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 126925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 126925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(126925\)    =    \(5^{2} \cdot 5077\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{126925} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 126925,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $14.85353048$
$L(\frac12)$  $\approx$  $14.85353048$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;5077\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;5077\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
5077 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−13.71610885583177, −13.12598093892908, −12.74225383383533, −12.52021762274178, −11.68701666641250, −10.98828885036076, −10.73748794323237, −10.40277845637787, −9.313636518192529, −9.082624201888768, −8.431210386273173, −8.142522081606284, −7.604357082046395, −7.290892714872026, −6.488002639970026, −5.661315083700317, −5.399727119550133, −4.760928579711444, −4.277340241362241, −3.767771082641743, −3.293616705756123, −2.611686409205161, −2.238503025342060, −1.739659150213529, −0.7866958165294847, 0.7866958165294847, 1.739659150213529, 2.238503025342060, 2.611686409205161, 3.293616705756123, 3.767771082641743, 4.277340241362241, 4.760928579711444, 5.399727119550133, 5.661315083700317, 6.488002639970026, 7.290892714872026, 7.604357082046395, 8.142522081606284, 8.431210386273173, 9.082624201888768, 9.313636518192529, 10.40277845637787, 10.73748794323237, 10.98828885036076, 11.68701666641250, 12.52021762274178, 12.74225383383533, 13.12598093892908, 13.71610885583177

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