Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 23 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 5-s − 3·7-s + 9-s − 4·11-s − 2·12-s − 15-s + 4·16-s − 3·17-s + 5·19-s + 2·20-s − 3·21-s + 23-s + 25-s + 27-s + 6·28-s − 29-s + 8·31-s − 4·33-s + 3·35-s − 2·36-s + 4·37-s − 6·41-s + 4·43-s + 8·44-s − 45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.258·15-s + 16-s − 0.727·17-s + 1.14·19-s + 0.447·20-s − 0.654·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.13·28-s − 0.185·29-s + 1.43·31-s − 0.696·33-s + 0.507·35-s − 1/3·36-s + 0.657·37-s − 0.937·41-s + 0.609·43-s + 1.20·44-s − 0.149·45-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10005\)    =    \(3 \cdot 5 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10005} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 10005,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−16.83933556450473, −16.25874902902652, −15.62869473666497, −15.39377605226625, −14.60105478549672, −13.88201240914562, −13.41433523795680, −13.08065237832338, −12.50156213526060, −11.85500715385217, −10.98889857675423, −10.27393690024541, −9.688212498124283, −9.409609161753552, −8.466023174974830, −8.182135288181352, −7.409463722517379, −6.767742999994972, −5.892066868807489, −5.123844612937545, −4.519762130975307, −3.696064517701593, −3.107351547206115, −2.456465682567805, −0.9582095215968572, 0, 0.9582095215968572, 2.456465682567805, 3.107351547206115, 3.696064517701593, 4.519762130975307, 5.123844612937545, 5.892066868807489, 6.767742999994972, 7.409463722517379, 8.182135288181352, 8.466023174974830, 9.409609161753552, 9.688212498124283, 10.27393690024541, 10.98889857675423, 11.85500715385217, 12.50156213526060, 13.08065237832338, 13.41433523795680, 13.88201240914562, 14.60105478549672, 15.39377605226625, 15.62869473666497, 16.25874902902652, 16.83933556450473

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