Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 83 $
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  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 15-s + 16-s − 3·17-s + 18-s + 6·19-s + 20-s − 2·22-s − 4·23-s + 24-s + 25-s + 27-s + 2·29-s + 30-s + 4·31-s + 32-s − 2·33-s − 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s − 0.348·33-s − 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(122010\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 83\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{122010} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 122010,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;83\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;83\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
83 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
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

−13.72031788391159, −13.57016846778654, −12.72375815611548, −12.62894058592475, −11.97414545521774, −11.34136628732865, −11.02777995131717, −10.33670716607667, −9.930713263109784, −9.409884881289140, −9.029227377008801, −8.111618848587964, −7.976910836225863, −7.366953287314790, −6.745882844965741, −6.238729309582668, −5.771157114034073, −5.092975071532689, −4.692862890237196, −4.095615171454185, −3.466415071512578, −2.719371258424510, −2.606591018088413, −1.686485473484713, −1.151596581729287, 0, 1.151596581729287, 1.686485473484713, 2.606591018088413, 2.719371258424510, 3.466415071512578, 4.095615171454185, 4.692862890237196, 5.092975071532689, 5.771157114034073, 6.238729309582668, 6.745882844965741, 7.366953287314790, 7.976910836225863, 8.111618848587964, 9.029227377008801, 9.409884881289140, 9.930713263109784, 10.33670716607667, 11.02777995131717, 11.34136628732865, 11.97414545521774, 12.62894058592475, 12.72375815611548, 13.57016846778654, 13.72031788391159

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