Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.901 - 0.432i$
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 + (−1.18 − 1.26i)3-s + (0.499 + 0.866i)4-s − 5-s + (0.389 + 1.68i)6-s + (2.05 + 1.66i)7-s − 0.999i·8-s + (−0.208 + 2.99i)9-s + (0.866 + 0.5i)10-s − 3.95i·11-s + (0.506 − 1.65i)12-s + (−2.82 − 1.63i)13-s + (−0.953 − 2.46i)14-s + (1.18 + 1.26i)15-s + (−0.5 + 0.866i)16-s + (0.497 − 0.861i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.682 − 0.731i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.159 + 0.688i)6-s + (0.778 + 0.627i)7-s − 0.353i·8-s + (−0.0695 + 0.997i)9-s + (0.273 + 0.158i)10-s − 1.19i·11-s + (0.146 − 0.478i)12-s + (−0.784 − 0.452i)13-s + (−0.254 − 0.659i)14-s + (0.305 + 0.327i)15-s + (−0.125 + 0.216i)16-s + (0.120 − 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.901 - 0.432i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.901 - 0.432i)\)
\(L(1)\)  \(\approx\)  \(0.0424560 + 0.186769i\)
\(L(\frac12)\)  \(\approx\)  \(0.0424560 + 0.186769i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.18 + 1.26i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.05 - 1.66i)T \)
good11 \( 1 + 3.95iT - 11T^{2} \)
13 \( 1 + (2.82 + 1.63i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.497 + 0.861i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.90 - 2.83i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.70iT - 23T^{2} \)
29 \( 1 + (4.10 - 2.36i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.83 - 2.79i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.05 + 8.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.16 - 8.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.10 - 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.36 - 4.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.82 + 5.09i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.70 + 2.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.45 - 3.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.61 + 7.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.74iT - 71T^{2} \)
73 \( 1 + (12.7 + 7.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.920 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.789 - 1.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.92 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.776 + 0.448i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.47263646618846963143085088771, −9.032305394854822588181918099575, −8.235194887207790520938672684773, −7.71599558665961247548262129652, −6.56980807999092603581673197791, −5.64668919060885285100716963243, −4.59176903699473750945409951797, −2.95666542392445010331471746152, −1.73751610796011111804761214891, −0.13581194752278408675600148424, 1.82516018646147424949308923425, 3.88276165077400000033567135196, 4.69567393366450342296656412070, 5.54552385213357130038673273905, 6.99040145179908378635498213743, 7.33965507956387850758624713930, 8.563049897748026256364565286324, 9.488813570338783833481741371785, 10.18575651577465252104577611168, 10.95783551839242859942526769129

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