Properties

Degree 2
Conductor $ 2^{3} \cdot 17 \cdot 59 $
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  − 0.587·3-s + 0.0220·5-s + 3.20·7-s − 2.65·9-s − 2.26·11-s + 0.668·13-s − 0.0129·15-s − 17-s + 4.04·19-s − 1.88·21-s − 0.862·23-s − 4.99·25-s + 3.32·27-s + 3.45·29-s − 1.93·31-s + 1.33·33-s + 0.0707·35-s − 0.186·37-s − 0.392·39-s + 4.18·41-s − 12.0·43-s − 0.0584·45-s + 1.63·47-s + 3.29·49-s + 0.587·51-s − 2.83·53-s − 0.0499·55-s + ⋯
L(s)  = 1  − 0.339·3-s + 0.00985·5-s + 1.21·7-s − 0.884·9-s − 0.683·11-s + 0.185·13-s − 0.00334·15-s − 0.242·17-s + 0.928·19-s − 0.411·21-s − 0.179·23-s − 0.999·25-s + 0.639·27-s + 0.641·29-s − 0.347·31-s + 0.231·33-s + 0.0119·35-s − 0.0305·37-s − 0.0629·39-s + 0.653·41-s − 1.83·43-s − 0.00871·45-s + 0.238·47-s + 0.471·49-s + 0.0822·51-s − 0.389·53-s − 0.00673·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8024,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 + T \)
good3 \( 1 + 0.587T + 3T^{2} \)
5 \( 1 - 0.0220T + 5T^{2} \)
7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
13 \( 1 - 0.668T + 13T^{2} \)
19 \( 1 - 4.04T + 19T^{2} \)
23 \( 1 + 0.862T + 23T^{2} \)
29 \( 1 - 3.45T + 29T^{2} \)
31 \( 1 + 1.93T + 31T^{2} \)
37 \( 1 + 0.186T + 37T^{2} \)
41 \( 1 - 4.18T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 - 1.63T + 47T^{2} \)
53 \( 1 + 2.83T + 53T^{2} \)
61 \( 1 + 3.86T + 61T^{2} \)
67 \( 1 - 9.45T + 67T^{2} \)
71 \( 1 - 8.22T + 71T^{2} \)
73 \( 1 + 4.56T + 73T^{2} \)
79 \( 1 + 7.12T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 7.29T + 97T^{2} \)
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

−7.69156556155728641907767071419, −6.76948272913019431902181898121, −5.97865941977165301906748138813, −5.28100672142042335106258563832, −4.93022406050121589311433291248, −3.96403349363284583551167115758, −3.03716447569858788568912281784, −2.20363270162820358672655191664, −1.26100012301555654582753442500, 0, 1.26100012301555654582753442500, 2.20363270162820358672655191664, 3.03716447569858788568912281784, 3.96403349363284583551167115758, 4.93022406050121589311433291248, 5.28100672142042335106258563832, 5.97865941977165301906748138813, 6.76948272913019431902181898121, 7.69156556155728641907767071419

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