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  − 2.07·3-s − 3.30·5-s + 0.766·7-s + 1.29·9-s − 4.04·11-s − 3.72·13-s + 6.84·15-s − 17-s + 0.733·19-s − 1.58·21-s − 1.80·23-s + 5.90·25-s + 3.52·27-s + 2.90·29-s + 1.23·31-s + 8.37·33-s − 2.53·35-s − 6.26·37-s + 7.72·39-s + 7.97·41-s + 0.613·43-s − 4.28·45-s + 13.3·47-s − 6.41·49-s + 2.07·51-s + 0.757·53-s + 13.3·55-s + ⋯
L(s)  = 1  − 1.19·3-s − 1.47·5-s + 0.289·7-s + 0.432·9-s − 1.21·11-s − 1.03·13-s + 1.76·15-s − 0.242·17-s + 0.168·19-s − 0.346·21-s − 0.376·23-s + 1.18·25-s + 0.679·27-s + 0.540·29-s + 0.221·31-s + 1.45·33-s − 0.427·35-s − 1.03·37-s + 1.23·39-s + 1.24·41-s + 0.0935·43-s − 0.638·45-s + 1.95·47-s − 0.916·49-s + 0.290·51-s + 0.104·53-s + 1.79·55-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 \( 1 + 2.07T + 3T^{2} \)
5 \( 1 + 3.30T + 5T^{2} \)
7 \( 1 - 0.766T + 7T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 + 3.72T + 13T^{2} \)
19 \( 1 - 0.733T + 19T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 1.23T + 31T^{2} \)
37 \( 1 + 6.26T + 37T^{2} \)
41 \( 1 - 7.97T + 41T^{2} \)
43 \( 1 - 0.613T + 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 - 0.757T + 53T^{2} \)
61 \( 1 - 4.13T + 61T^{2} \)
67 \( 1 + 9.80T + 67T^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 - 9.46T + 73T^{2} \)
79 \( 1 + 6.71T + 79T^{2} \)
83 \( 1 + 0.845T + 83T^{2} \)
89 \( 1 + 0.654T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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

−7.42769949142947304651624615302, −6.95671108302790526018188472548, −5.99127204715588889308024149472, −5.29697210546240100400794239664, −4.73677788816731435618693973829, −4.17220598013541935553147983276, −3.13320505616159000675255968505, −2.30431281641749004316734334960, −0.75239854780050374802032284045, 0, 0.75239854780050374802032284045, 2.30431281641749004316734334960, 3.13320505616159000675255968505, 4.17220598013541935553147983276, 4.73677788816731435618693973829, 5.29697210546240100400794239664, 5.99127204715588889308024149472, 6.95671108302790526018188472548, 7.42769949142947304651624615302

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