Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.744 + 0.667i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 + 0.390i)2-s + (1.69 − 1.06i)4-s − 1.82i·5-s + 7-s + (−1.88 + 2.10i)8-s + (0.714 + 2.48i)10-s + 3.75i·11-s − 3.09i·13-s + (−1.35 + 0.390i)14-s + (1.74 − 3.59i)16-s − 3.61·17-s − 1.65i·19-s + (−1.94 − 3.10i)20-s + (−1.46 − 5.10i)22-s + 7.47·23-s + ⋯
L(s)  = 1  + (−0.961 + 0.276i)2-s + (0.847 − 0.531i)4-s − 0.818i·5-s + 0.377·7-s + (−0.667 + 0.744i)8-s + (0.226 + 0.786i)10-s + 1.13i·11-s − 0.859i·13-s + (−0.363 + 0.104i)14-s + (0.436 − 0.899i)16-s − 0.875·17-s − 0.379i·19-s + (−0.434 − 0.693i)20-s + (−0.313 − 1.08i)22-s + 1.55·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.744 + 0.667i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.744 + 0.667i)$
$L(1)$  $\approx$  $1.088225265$
$L(\frac12)$  $\approx$  $1.088225265$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.390i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 1.82iT - 5T^{2} \)
11 \( 1 - 3.75iT - 11T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + 1.65iT - 19T^{2} \)
23 \( 1 - 7.47T + 23T^{2} \)
29 \( 1 + 2.38iT - 29T^{2} \)
31 \( 1 - 8.97T + 31T^{2} \)
37 \( 1 - 8.94iT - 37T^{2} \)
41 \( 1 + 3.04T + 41T^{2} \)
43 \( 1 + 1.82iT - 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 + 3.21iT - 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 + 9.91iT - 61T^{2} \)
67 \( 1 + 8.73iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 + 7.24T + 79T^{2} \)
83 \( 1 + 8.99iT - 83T^{2} \)
89 \( 1 - 1.54T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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

−9.398146106878940724690742589694, −8.422400228831728123005423349179, −8.101897372981374818170336474995, −6.99447931780057572151839169092, −6.44323647755374423350412545667, −4.98686208975378646763357636089, −4.83969570689036784762506456478, −3.01659388207804353367348239991, −1.87106891205974548028381574382, −0.71539293461486830415860667583, 1.08878820625158669613068739110, 2.43954113615490349982128930309, 3.20169606227025896144803337164, 4.32414259420699961591783454471, 5.69865444168771703435105551791, 6.72286343029555430061692094721, 7.04597160356929425994014125218, 8.206304446009929417661230740377, 8.758725623856991933062282431333, 9.473542266985926330798698515969

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