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 − 5·7-s + 9-s + 2·12-s − 4·13-s + 15-s + 4·16-s + 3·17-s − 7·19-s + 2·20-s + 5·21-s − 23-s + 25-s − 27-s + 10·28-s − 29-s − 8·31-s + 5·35-s − 2·36-s + 4·37-s + 4·39-s + 2·41-s + 8·43-s − 45-s + 6·47-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s + 0.727·17-s − 1.60·19-s + 0.447·20-s + 1.09·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.88·28-s − 0.185·29-s − 1.43·31-s + 0.845·35-s − 1/3·36-s + 0.657·37-s + 0.640·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
good2 \( 1 + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−16.96723896522345, −16.56025374118693, −15.80740538187951, −15.31360521911730, −14.56010980869474, −14.14518773003852, −13.13051550392778, −12.87951994910884, −12.39131854562031, −12.05475819285217, −10.83727095593822, −10.53077059782511, −9.649119436232221, −9.474251251994725, −8.801455112179655, −7.851313481328604, −7.326852644115285, −6.541389317380707, −5.943229245718141, −5.336298575707143, −4.423323678366348, −3.903632830734584, −3.220621302043720, −2.263290874296234, −0.6742042071137102, 0, 0.6742042071137102, 2.263290874296234, 3.220621302043720, 3.903632830734584, 4.423323678366348, 5.336298575707143, 5.943229245718141, 6.541389317380707, 7.326852644115285, 7.851313481328604, 8.801455112179655, 9.474251251994725, 9.649119436232221, 10.53077059782511, 10.83727095593822, 12.05475819285217, 12.39131854562031, 12.87951994910884, 13.13051550392778, 14.14518773003852, 14.56010980869474, 15.31360521911730, 15.80740538187951, 16.56025374118693, 16.96723896522345

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