Properties

Degree 2
Conductor $ 7^{2} \cdot 11 $
Sign $-0.979 + 0.200i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.116 − 1.10i)2-s + (1.91 − 2.12i)3-s + (0.741 − 0.157i)4-s + (−3.15 + 1.40i)5-s + (−2.57 − 1.87i)6-s + (−0.949 − 2.92i)8-s + (−0.540 − 5.14i)9-s + (1.92 + 3.33i)10-s + (−1.57 − 2.91i)11-s + (1.08 − 1.87i)12-s + (−1.66 + 1.21i)13-s + (−3.05 + 9.39i)15-s + (−1.74 + 0.776i)16-s + (0.202 − 1.92i)17-s + (−5.63 + 1.19i)18-s + (1.58 + 0.337i)19-s + ⋯
L(s)  = 1  + (−0.0823 − 0.783i)2-s + (1.10 − 1.22i)3-s + (0.370 − 0.0787i)4-s + (−1.41 + 0.628i)5-s + (−1.05 − 0.764i)6-s + (−0.335 − 1.03i)8-s + (−0.180 − 1.71i)9-s + (0.609 + 1.05i)10-s + (−0.475 − 0.879i)11-s + (0.312 − 0.541i)12-s + (−0.462 + 0.335i)13-s + (−0.788 + 2.42i)15-s + (−0.436 + 0.194i)16-s + (0.0490 − 0.466i)17-s + (−1.32 + 0.282i)18-s + (0.364 + 0.0774i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(539\)    =    \(7^{2} \cdot 11\)
\( \varepsilon \)  =  $-0.979 + 0.200i$
motivic weight  =  \(1\)
character  :  $\chi_{539} (520, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 539,\ (\ :1/2),\ -0.979 + 0.200i)$
$L(1)$  $\approx$  $0.159293 - 1.57641i$
$L(\frac12)$  $\approx$  $0.159293 - 1.57641i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + (1.57 + 2.91i)T \)
good2 \( 1 + (0.116 + 1.10i)T + (-1.95 + 0.415i)T^{2} \)
3 \( 1 + (-1.91 + 2.12i)T + (-0.313 - 2.98i)T^{2} \)
5 \( 1 + (3.15 - 1.40i)T + (3.34 - 3.71i)T^{2} \)
13 \( 1 + (1.66 - 1.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.202 + 1.92i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-1.58 - 0.337i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-0.403 + 0.698i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.46 + 7.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.720 + 0.320i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-6.73 - 7.47i)T + (-3.86 + 36.7i)T^{2} \)
41 \( 1 + (-0.657 - 2.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.08T + 43T^{2} \)
47 \( 1 + (-7.40 - 1.57i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (-9.88 - 4.40i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (3.22 - 0.685i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-0.983 + 0.437i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (1.20 + 2.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.57 - 1.87i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.19 + 0.254i)T + (66.6 - 29.6i)T^{2} \)
79 \( 1 + (-0.991 - 9.42i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (13.0 + 9.44i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.21 - 3.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.23 + 3.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.62402875427462712547749356558, −9.521769918736782544651245949983, −8.404330148844244296877522989843, −7.67092662772629204634622373534, −7.13442252245000385464891265038, −6.18498230139203127179382512523, −4.08044240914432014971524488448, −3.00994163651349475458413033418, −2.52788499021341550431357972744, −0.824748616994485156491565400559, 2.53462794725966574450429495515, 3.65705684753398484773324249163, 4.57643213949137292229356083305, 5.41289930593568522994862903211, 7.22485756649214795914994700608, 7.70963663962358651007740808146, 8.502361031417907300329125495672, 9.132129668642173726763290998490, 10.26210030664498806849586154049, 11.04807409664997113666973227166

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