Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.231 + 0.972i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.189 + 2.63i)7-s − 0.999i·8-s + (−3.44 − 1.99i)11-s − 0.0681i·13-s + (1.48 + 2.19i)14-s + (−0.5 − 0.866i)16-s + (3.66 − 6.34i)17-s + (−1.76 + 1.01i)19-s − 3.98·22-s + (3.23 − 1.86i)23-s + (−0.0340 − 0.0590i)26-s + (2.38 + 1.15i)28-s − 0.898i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0716 + 0.997i)7-s − 0.353i·8-s + (−1.03 − 0.600i)11-s − 0.0189i·13-s + (0.396 + 0.585i)14-s + (−0.125 − 0.216i)16-s + (0.888 − 1.53i)17-s + (−0.404 + 0.233i)19-s − 0.848·22-s + (0.673 − 0.389i)23-s + (−0.00668 − 0.0115i)26-s + (0.449 + 0.218i)28-s − 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.231 + 0.972i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.231 + 0.972i)$
$L(1)$  $\approx$  $2.064863401$
$L(\frac12)$  $\approx$  $2.064863401$
$L(\frac{3}{2})$   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 \( 1 + (-0.189 - 2.63i)T \)
good11 \( 1 + (3.44 + 1.99i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.0681iT - 13T^{2} \)
17 \( 1 + (-3.66 + 6.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.76 - 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.03 + 3.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 - 0.964T + 43T^{2} \)
47 \( 1 + (0.830 + 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.4 - 6.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.32 + 9.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 + 3.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.33 + 9.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.93iT - 71T^{2} \)
73 \( 1 + (-10.0 - 5.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.77 + 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + (-0.913 - 1.58i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.513178077045828891619870169055, −7.70271794751404502967465823207, −6.91332342987154538737805517765, −5.87137490991587160451956133377, −5.37721181833880787767519093182, −4.78294358216552204630368938013, −3.53345289779169516689751430260, −2.80820857365368323887647466723, −2.07947607705290411439155733421, −0.51726468778802505605677853855, 1.30803123899090426892365242905, 2.50373110256644709819093863215, 3.60264747695957615758548139880, 4.16705907898217585014263173171, 5.17071540270953915265224493700, 5.66922254760542994691942545394, 6.86046111823380545474430600620, 7.17763723818131033961312670177, 8.088944855763630663897283292447, 8.558420740612561511484931648337

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