Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·9-s + 2·11-s − 15-s + 4·17-s − 2·19-s + 23-s + 25-s − 5·27-s + 9·29-s + 4·31-s + 2·33-s + 4·37-s + 41-s + 9·43-s + 2·45-s + 4·51-s − 10·53-s − 2·55-s − 2·57-s − 10·59-s + 9·61-s + 5·67-s + 69-s + 14·71-s + 12·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.603·11-s − 0.258·15-s + 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.962·27-s + 1.67·29-s + 0.718·31-s + 0.348·33-s + 0.657·37-s + 0.156·41-s + 1.37·43-s + 0.298·45-s + 0.560·51-s − 1.37·53-s − 0.269·55-s − 0.264·57-s − 1.30·59-s + 1.15·61-s + 0.610·67-s + 0.120·69-s + 1.66·71-s + 1.40·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1960} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1960,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.952924545$
$L(\frac12)$  $\approx$  $1.952924545$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−19.38611625744626, −19.11875487618814, −18.18520843938885, −17.38256889257185, −16.90649900373131, −16.14389191896248, −15.41457818132586, −14.79952018867264, −14.06554212182831, −13.83661836589777, −12.57072988822706, −12.23223835150143, −11.33636542672518, −10.75926217341491, −9.720054458442191, −9.184485842190697, −8.149986311078948, −8.018804392661813, −6.804823593989008, −6.111062415576410, −5.075797997202586, −4.117805760621083, −3.258010881705971, −2.432158504141342, −0.9317825218195481, 0.9317825218195481, 2.432158504141342, 3.258010881705971, 4.117805760621083, 5.075797997202586, 6.111062415576410, 6.804823593989008, 8.018804392661813, 8.149986311078948, 9.184485842190697, 9.720054458442191, 10.75926217341491, 11.33636542672518, 12.23223835150143, 12.57072988822706, 13.83661836589777, 14.06554212182831, 14.79952018867264, 15.41457818132586, 16.14389191896248, 16.90649900373131, 17.38256889257185, 18.18520843938885, 19.11875487618814, 19.38611625744626

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