Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.994 - 0.106i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.994 - 0.106i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.994 - 0.106i)$
$L(1)$  $\approx$  $0.6441563695$
$L(\frac12)$  $\approx$  $0.6441563695$
$L(\frac{3}{2})$   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

−8.361714836155389265909679097529, −7.63057236137145779780241213639, −6.78514732823847913345642033920, −6.42148197021308749698301517316, −5.07200911445211888480071256853, −4.22217894602866376809715187188, −3.49661915373879796242190744178, −2.45130708644196121041914425772, −1.28990333862767017115949858069, −0.24787602332205104803329279802, 1.63796282448156504885092077764, 2.17610158209895862146298291427, 3.62359060013966794349398271639, 4.43184050141430195996966375174, 5.50249596938278391835568062688, 6.21671179958968555310223473039, 6.70386104831711803395130288292, 7.73797180154625839384052925688, 8.374315958718254232322473023901, 9.072505620016497673986577896243

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