Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 151 $
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  − 1.57·3-s + 5-s − 0.208·7-s − 0.513·9-s − 0.616·11-s − 3.24·13-s − 1.57·15-s + 1.94·17-s + 1.81·19-s + 0.329·21-s − 1.42·23-s + 25-s + 5.54·27-s − 4.26·29-s + 6.21·31-s + 0.971·33-s − 0.208·35-s + 4.31·37-s + 5.11·39-s + 1.53·41-s − 1.84·43-s − 0.513·45-s − 4.09·47-s − 6.95·49-s − 3.07·51-s + 10.8·53-s − 0.616·55-s + ⋯
L(s)  = 1  − 0.910·3-s + 0.447·5-s − 0.0789·7-s − 0.171·9-s − 0.185·11-s − 0.899·13-s − 0.407·15-s + 0.472·17-s + 0.416·19-s + 0.0718·21-s − 0.297·23-s + 0.200·25-s + 1.06·27-s − 0.792·29-s + 1.11·31-s + 0.169·33-s − 0.0352·35-s + 0.709·37-s + 0.818·39-s + 0.239·41-s − 0.281·43-s − 0.0766·45-s − 0.597·47-s − 0.993·49-s − 0.430·51-s + 1.48·53-s − 0.0831·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6040} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6040,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;151\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;151\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 - T \)
151 \( 1 + T \)
good3 \( 1 + 1.57T + 3T^{2} \)
7 \( 1 + 0.208T + 7T^{2} \)
11 \( 1 + 0.616T + 11T^{2} \)
13 \( 1 + 3.24T + 13T^{2} \)
17 \( 1 - 1.94T + 17T^{2} \)
19 \( 1 - 1.81T + 19T^{2} \)
23 \( 1 + 1.42T + 23T^{2} \)
29 \( 1 + 4.26T + 29T^{2} \)
31 \( 1 - 6.21T + 31T^{2} \)
37 \( 1 - 4.31T + 37T^{2} \)
41 \( 1 - 1.53T + 41T^{2} \)
43 \( 1 + 1.84T + 43T^{2} \)
47 \( 1 + 4.09T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 8.06T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 5.24T + 67T^{2} \)
71 \( 1 - 8.10T + 71T^{2} \)
73 \( 1 + 1.11T + 73T^{2} \)
79 \( 1 + 7.47T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 9.40T + 97T^{2} \)
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

−7.65869279783766180312126302724, −6.88552778896417701341207069578, −6.21180055267583860307963938683, −5.53669158793204083770866471388, −5.06834095802311079871838034949, −4.23101785026797301155178797918, −3.10168230642286775285003098760, −2.36037426181602344169641644563, −1.15099172947200930871463421380, 0, 1.15099172947200930871463421380, 2.36037426181602344169641644563, 3.10168230642286775285003098760, 4.23101785026797301155178797918, 5.06834095802311079871838034949, 5.53669158793204083770866471388, 6.21180055267583860307963938683, 6.88552778896417701341207069578, 7.65869279783766180312126302724

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