Properties

Degree 2
Conductor $ 5^{2} \cdot 7 \cdot 23 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 7-s − 3·8-s − 3·9-s − 4·11-s + 2·13-s + 14-s − 16-s − 6·17-s − 3·18-s + 4·19-s − 4·22-s − 23-s + 2·26-s − 28-s − 2·29-s + 4·31-s + 5·32-s − 6·34-s + 3·36-s + 10·37-s + 4·38-s + 10·41-s − 12·43-s + 4·44-s − 46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s − 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.852·22-s − 0.208·23-s + 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.883·32-s − 1.02·34-s + 1/2·36-s + 1.64·37-s + 0.648·38-s + 1.56·41-s − 1.82·43-s + 0.603·44-s − 0.147·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.544988028$
$L(\frac12)$  $\approx$  $1.544988028$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−8.257751522950231816488875233294, −8.016124128018196443937064440201, −6.80495531146416711043233780876, −5.94221101431227801078388923498, −5.42521227608983565445028389193, −4.71901779676615522584981056956, −3.97527841357547486453795945979, −2.97933848611012499979092091966, −2.34495592904849279164106500380, −0.61612818314657448403323002463, 0.61612818314657448403323002463, 2.34495592904849279164106500380, 2.97933848611012499979092091966, 3.97527841357547486453795945979, 4.71901779676615522584981056956, 5.42521227608983565445028389193, 5.94221101431227801078388923498, 6.80495531146416711043233780876, 8.016124128018196443937064440201, 8.257751522950231816488875233294

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