Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
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  − 1.83·2-s + 3-s + 1.38·4-s + 0.446·5-s − 1.83·6-s + 3.21·7-s + 1.13·8-s + 9-s − 0.821·10-s + 0.755·11-s + 1.38·12-s − 6.00·13-s − 5.90·14-s + 0.446·15-s − 4.85·16-s − 17-s − 1.83·18-s − 3.47·19-s + 0.617·20-s + 3.21·21-s − 1.38·22-s − 1.33·23-s + 1.13·24-s − 4.80·25-s + 11.0·26-s + 27-s + 4.44·28-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.577·3-s + 0.692·4-s + 0.199·5-s − 0.751·6-s + 1.21·7-s + 0.400·8-s + 0.333·9-s − 0.259·10-s + 0.227·11-s + 0.399·12-s − 1.66·13-s − 1.57·14-s + 0.115·15-s − 1.21·16-s − 0.242·17-s − 0.433·18-s − 0.797·19-s + 0.138·20-s + 0.700·21-s − 0.296·22-s − 0.278·23-s + 0.231·24-s − 0.960·25-s + 2.16·26-s + 0.192·27-s + 0.839·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 1.83T + 2T^{2} \)
5 \( 1 - 0.446T + 5T^{2} \)
7 \( 1 - 3.21T + 7T^{2} \)
11 \( 1 - 0.755T + 11T^{2} \)
13 \( 1 + 6.00T + 13T^{2} \)
19 \( 1 + 3.47T + 19T^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 + 5.99T + 29T^{2} \)
31 \( 1 - 0.937T + 31T^{2} \)
37 \( 1 - 7.61T + 37T^{2} \)
41 \( 1 - 4.83T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 - 4.65T + 47T^{2} \)
53 \( 1 + 0.522T + 53T^{2} \)
59 \( 1 + 3.59T + 59T^{2} \)
61 \( 1 + 3.89T + 61T^{2} \)
67 \( 1 + 4.41T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 - 3.70T + 83T^{2} \)
89 \( 1 - 1.35T + 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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

−7.56931560933043693148899571319, −7.41505802358419792773969132713, −6.30789579479806921501052756579, −5.39041617516468384171748731186, −4.45644760742346259025209058768, −4.14773461136695656995704207785, −2.53019979442147662696001215694, −2.13317599650542837055989698963, −1.27015701863682697104644227239, 0, 1.27015701863682697104644227239, 2.13317599650542837055989698963, 2.53019979442147662696001215694, 4.14773461136695656995704207785, 4.45644760742346259025209058768, 5.39041617516468384171748731186, 6.30789579479806921501052756579, 7.41505802358419792773969132713, 7.56931560933043693148899571319

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