Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 601 $
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 − 0.706·3-s + 4-s + 5-s − 0.706·6-s − 2.05·7-s + 8-s − 2.50·9-s + 10-s + 5.00·11-s − 0.706·12-s + 1.44·13-s − 2.05·14-s − 0.706·15-s + 16-s − 5.11·17-s − 2.50·18-s − 0.185·19-s + 20-s + 1.45·21-s + 5.00·22-s − 2.44·23-s − 0.706·24-s + 25-s + 1.44·26-s + 3.88·27-s − 2.05·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.408·3-s + 0.5·4-s + 0.447·5-s − 0.288·6-s − 0.776·7-s + 0.353·8-s − 0.833·9-s + 0.316·10-s + 1.50·11-s − 0.204·12-s + 0.400·13-s − 0.548·14-s − 0.182·15-s + 0.250·16-s − 1.23·17-s − 0.589·18-s − 0.0424·19-s + 0.223·20-s + 0.316·21-s + 1.06·22-s − 0.510·23-s − 0.144·24-s + 0.200·25-s + 0.283·26-s + 0.748·27-s − 0.388·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6010\)    =    \(2 \cdot 5 \cdot 601\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;601\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;601\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
601 \( 1 + T \)
good3 \( 1 + 0.706T + 3T^{2} \)
7 \( 1 + 2.05T + 7T^{2} \)
11 \( 1 - 5.00T + 11T^{2} \)
13 \( 1 - 1.44T + 13T^{2} \)
17 \( 1 + 5.11T + 17T^{2} \)
19 \( 1 + 0.185T + 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 + 2.29T + 29T^{2} \)
31 \( 1 + 9.99T + 31T^{2} \)
37 \( 1 + 5.58T + 37T^{2} \)
41 \( 1 - 2.67T + 41T^{2} \)
43 \( 1 - 2.86T + 43T^{2} \)
47 \( 1 + 1.49T + 47T^{2} \)
53 \( 1 - 9.86T + 53T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 - 1.70T + 61T^{2} \)
67 \( 1 + 4.31T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 1.68T + 73T^{2} \)
79 \( 1 - 3.43T + 79T^{2} \)
83 \( 1 - 0.478T + 83T^{2} \)
89 \( 1 + 7.20T + 89T^{2} \)
97 \( 1 + 4.23T + 97T^{2} \)
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

−7.39752103712218294510726889245, −6.75904749888775689243314093667, −6.12134533560762655205904299130, −5.82260013715011014113253056548, −4.86683994903485833201975284267, −3.96461290223578799871802507222, −3.42496660568190813218035762307, −2.40389098770670513202946977181, −1.49374951023143691191529119945, 0, 1.49374951023143691191529119945, 2.40389098770670513202946977181, 3.42496660568190813218035762307, 3.96461290223578799871802507222, 4.86683994903485833201975284267, 5.82260013715011014113253056548, 6.12134533560762655205904299130, 6.75904749888775689243314093667, 7.39752103712218294510726889245

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