Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.564 - 0.825i$
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 + (−1.71 − 0.219i)3-s + (0.499 + 0.866i)4-s − 5-s + (1.37 + 1.04i)6-s + (−2.11 − 1.59i)7-s − 0.999i·8-s + (2.90 + 0.755i)9-s + (0.866 + 0.5i)10-s − 0.767i·11-s + (−0.668 − 1.59i)12-s + (−3.78 − 2.18i)13-s + (1.03 + 2.43i)14-s + (1.71 + 0.219i)15-s + (−0.5 + 0.866i)16-s + (−1.15 + 1.99i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.991 − 0.126i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.562 + 0.428i)6-s + (−0.798 − 0.602i)7-s − 0.353i·8-s + (0.967 + 0.251i)9-s + (0.273 + 0.158i)10-s − 0.231i·11-s + (−0.193 − 0.461i)12-s + (−1.04 − 0.605i)13-s + (0.276 + 0.650i)14-s + (0.443 + 0.0567i)15-s + (−0.125 + 0.216i)16-s + (−0.279 + 0.483i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.564 - 0.825i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.564 - 0.825i)\)
\(L(1)\)  \(\approx\)  \(0.314181 + 0.165880i\)
\(L(\frac12)\)  \(\approx\)  \(0.314181 + 0.165880i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.71 + 0.219i)T \)
5 \( 1 + T \)
7 \( 1 + (2.11 + 1.59i)T \)
good11 \( 1 + 0.767iT - 11T^{2} \)
13 \( 1 + (3.78 + 2.18i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.15 - 1.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.19 + 2.42i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.59iT - 23T^{2} \)
29 \( 1 + (9.12 - 5.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.97 + 4.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.15 - 7.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.65 - 4.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.94 - 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.23 + 5.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.48 - 3.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.71 - 9.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.22 + 4.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.85 + 11.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.50iT - 71T^{2} \)
73 \( 1 + (-10.1 - 5.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.555 + 0.961i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.254 + 0.440i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.80 - 4.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.61 - 4.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.76538343341283710219441596673, −9.897579912826412855401168818128, −9.368107492251895837280801756932, −7.82294635511170190189134632958, −7.34433150656808262056752841361, −6.39831139578994252781437121804, −5.29159802150006246508583675802, −4.10505606191829646647662481270, −2.93412298893514027548704184662, −1.04410207723864341761084264740, 0.33046425434303248010718232347, 2.34244793313910316238607750400, 4.03598658119313170245729588678, 5.16730859515705063210813565240, 6.02275680543794540764955178011, 6.97800106672283292865735693388, 7.55227345288329282928138089247, 8.936668132427823110491094368917, 9.625623778625140848109071558639, 10.30294054396613137900010592877

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