Properties

Label 2-840-1.1-c1-0-0
Degree $2$
Conductor $840$
Sign $1$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 2·13-s + 15-s + 6·17-s − 4·19-s − 21-s + 4·23-s + 25-s − 27-s + 6·29-s − 35-s + 6·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s + 8·47-s + 49-s − 6·51-s + 14·53-s + 4·57-s − 4·59-s − 2·61-s + 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.169·35-s + 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.92·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.221288344\)
\(L(\frac12)\) \(\approx\) \(1.221288344\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35899536139237859657452871452, −9.450754894645541234345596216334, −8.393189724041127383030805741896, −7.62422934296928339340232971391, −6.79350883637272350613856089385, −5.74083335944297837978899468723, −4.88159926035910944720765525241, −3.96302542474933435088975798400, −2.62319993876382304656802984913, −0.954082357122462532643032272715, 0.954082357122462532643032272715, 2.62319993876382304656802984913, 3.96302542474933435088975798400, 4.88159926035910944720765525241, 5.74083335944297837978899468723, 6.79350883637272350613856089385, 7.62422934296928339340232971391, 8.393189724041127383030805741896, 9.450754894645541234345596216334, 10.35899536139237859657452871452

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