Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 19 \cdot 47 $
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 − 3·5-s − 7-s − 2·9-s + 4·13-s + 3·15-s + 4·17-s + 19-s + 21-s + 8·23-s + 4·25-s + 5·27-s + 5·29-s + 2·31-s + 3·35-s + 4·37-s − 4·39-s − 2·41-s + 12·43-s + 6·45-s − 47-s + 49-s − 4·51-s + 4·53-s − 57-s − 12·59-s + 12·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s + 1.10·13-s + 0.774·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s + 1.66·23-s + 4/5·25-s + 0.962·27-s + 0.928·29-s + 0.359·31-s + 0.507·35-s + 0.657·37-s − 0.640·39-s − 0.312·41-s + 1.82·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s − 0.560·51-s + 0.549·53-s − 0.132·57-s − 1.56·59-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100016\)    =    \(2^{4} \cdot 7 \cdot 19 \cdot 47\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100016} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100016,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;19,\;47\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;19,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
19 \( 1 - T \)
47 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−14.06988623534651, −13.48501750451015, −12.86349627754919, −12.43209577791246, −11.99175322480443, −11.55924311055584, −11.00949178427018, −10.88465710449639, −10.19167248546091, −9.520867661519705, −8.907986747452354, −8.532459247840279, −7.942412052242811, −7.597588222493320, −6.789436748277248, −6.558172005641830, −5.720250577484182, −5.454740944409244, −4.689061080779858, −4.145996965901732, −3.561674445364770, −3.016547624268454, −2.602161051966896, −1.130907371183474, −0.9197513854996921, 0, 0.9197513854996921, 1.130907371183474, 2.602161051966896, 3.016547624268454, 3.561674445364770, 4.145996965901732, 4.689061080779858, 5.454740944409244, 5.720250577484182, 6.558172005641830, 6.789436748277248, 7.597588222493320, 7.942412052242811, 8.532459247840279, 8.907986747452354, 9.520867661519705, 10.19167248546091, 10.88465710449639, 11.00949178427018, 11.55924311055584, 11.99175322480443, 12.43209577791246, 12.86349627754919, 13.48501750451015, 14.06988623534651

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