Properties

Degree 2
Conductor $ 2^{2} \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  − 3-s + 4.39·5-s + 7-s − 2·9-s − 2·11-s − 3.39·13-s − 4.39·15-s + 17-s − 6.39·19-s − 21-s + 0.222·23-s + 14.3·25-s + 5·27-s + 2.39·29-s − 6.57·31-s + 2·33-s + 4.39·35-s − 9.97·37-s + 3.39·39-s − 6.39·41-s − 10.5·43-s − 8.79·45-s − 4.22·47-s − 6·49-s − 51-s + 13.3·53-s − 8.79·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.96·5-s + 0.377·7-s − 0.666·9-s − 0.603·11-s − 0.942·13-s − 1.13·15-s + 0.242·17-s − 1.46·19-s − 0.218·21-s + 0.0464·23-s + 2.87·25-s + 0.962·27-s + 0.445·29-s − 1.18·31-s + 0.348·33-s + 0.743·35-s − 1.63·37-s + 0.544·39-s − 0.999·41-s − 1.61·43-s − 1.31·45-s − 0.615·47-s − 0.857·49-s − 0.140·51-s + 1.83·53-s − 1.18·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T + 3T^{2} \)
5 \( 1 - 4.39T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 3.39T + 13T^{2} \)
19 \( 1 + 6.39T + 19T^{2} \)
23 \( 1 - 0.222T + 23T^{2} \)
29 \( 1 - 2.39T + 29T^{2} \)
31 \( 1 + 6.57T + 31T^{2} \)
37 \( 1 + 9.97T + 37T^{2} \)
41 \( 1 + 6.39T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 4.22T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
61 \( 1 - 1.39T + 61T^{2} \)
67 \( 1 + 2.44T + 67T^{2} \)
71 \( 1 - 6.35T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + 4.57T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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

−8.387536970408179865519597184641, −6.99461622700724687408939768802, −6.59260303077055468452134603944, −5.63128930390334481218375890573, −5.32492384863894211487470495345, −4.68861124239776947928216451303, −3.15291237176619664944276582847, −2.27721176831537020118005293547, −1.64597547128874264796108915571, 0, 1.64597547128874264796108915571, 2.27721176831537020118005293547, 3.15291237176619664944276582847, 4.68861124239776947928216451303, 5.32492384863894211487470495345, 5.63128930390334481218375890573, 6.59260303077055468452134603944, 6.99461622700724687408939768802, 8.387536970408179865519597184641

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