Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.705 + 0.708i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (2.34 + 1.22i)7-s − 0.999i·8-s + (2.03 + 1.17i)11-s − 4.64i·13-s + (2.64 − 0.107i)14-s + (−0.5 − 0.866i)16-s + (−2.28 + 3.95i)17-s + (0.491 − 0.283i)19-s + 2.35·22-s + (5.04 − 2.91i)23-s + (−2.32 − 4.02i)26-s + (2.23 − 1.41i)28-s − 2.55i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.885 + 0.464i)7-s − 0.353i·8-s + (0.615 + 0.355i)11-s − 1.28i·13-s + (0.706 − 0.0287i)14-s + (−0.125 − 0.216i)16-s + (−0.553 + 0.958i)17-s + (0.112 − 0.0650i)19-s + 0.502·22-s + (1.05 − 0.607i)23-s + (−0.455 − 0.789i)26-s + (0.422 − 0.267i)28-s − 0.474i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.705 + 0.708i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.705 + 0.708i)$
$L(1)$  $\approx$  $3.177927050$
$L(\frac12)$  $\approx$  $3.177927050$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.34 - 1.22i)T \)
good11 \( 1 + (-2.03 - 1.17i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.64iT - 13T^{2} \)
17 \( 1 + (2.28 - 3.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.491 + 0.283i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.04 + 2.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.63 + 8.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 - 6.57T + 43T^{2} \)
47 \( 1 + (-3.15 - 5.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 6.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.67 + 2.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.85 - 3.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.00 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.02iT - 71T^{2} \)
73 \( 1 + (7.11 + 4.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.13 - 7.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.171T + 83T^{2} \)
89 \( 1 + (2.72 + 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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

−8.730901496082647157888909113755, −7.75173876138902512185923493467, −7.13553852893532735878815792107, −5.96866467998547943495608532785, −5.61341098735552898070788663467, −4.58173645591170714487262958108, −4.02712450805482014289543777533, −2.87263111054360904000098397416, −2.07854064124657217364919855209, −0.952507146783937178262348253080, 1.15769551432101906697641085850, 2.23863401983396724805896293634, 3.40640453091644541147186319662, 4.24514275232284722371712140867, 4.85969734489993244424714457465, 5.61876625987502422858987467935, 6.72022467783895115837833436862, 7.07361539265510294102208366460, 7.83808208993057779703905695375, 8.959654314683990176296792240099

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