Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.973 + 0.229i$
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 + (−1.22 + 1.22i)3-s + (0.499 − 0.866i)4-s − 5-s + (−0.443 + 1.67i)6-s + (2.64 − 0.169i)7-s − 0.999i·8-s + (−0.0155 − 2.99i)9-s + (−0.866 + 0.5i)10-s − 0.207i·11-s + (0.452 + 1.67i)12-s + (1.22 − 0.707i)13-s + (2.20 − 1.46i)14-s + (1.22 − 1.22i)15-s + (−0.5 − 0.866i)16-s + (0.680 + 1.17i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.705 + 0.708i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.181 + 0.683i)6-s + (0.997 − 0.0641i)7-s − 0.353i·8-s + (−0.00516 − 0.999i)9-s + (−0.273 + 0.158i)10-s − 0.0626i·11-s + (0.130 + 0.482i)12-s + (0.339 − 0.196i)13-s + (0.588 − 0.392i)14-s + (0.315 − 0.317i)15-s + (−0.125 − 0.216i)16-s + (0.165 + 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.973 + 0.229i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (551, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.973 + 0.229i)\)
\(L(1)\)  \(\approx\)  \(1.76822 - 0.205968i\)
\(L(\frac12)\)  \(\approx\)  \(1.76822 - 0.205968i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.64 + 0.169i)T \)
good11 \( 1 + 0.207iT - 11T^{2} \)
13 \( 1 + (-1.22 + 0.707i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.680 - 1.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.73 - 3.30i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.39iT - 23T^{2} \)
29 \( 1 + (-6.11 - 3.53i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.10 - 2.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.32 + 2.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.41 + 2.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.06 + 1.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0573 + 0.0993i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.40 - 4.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.34 + 7.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.85 - 1.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0235 - 0.0407i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.45iT - 71T^{2} \)
73 \( 1 + (11.5 - 6.68i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.97 + 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.06 - 13.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.40 - 2.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.39 - 0.803i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.63634549881443640709123833242, −10.14853209818691014438585515332, −8.887129422588452494615631950323, −7.967783460811675353853113643545, −6.76797177181280140381639121034, −5.72915377655300769020148083487, −4.90096573122683405557805918124, −4.14627005503824215432420210274, −3.06692470789502806593177874355, −1.15422805191874139373068366268, 1.28932493440538180621631624753, 2.86456833167719857601032116591, 4.41136007467401089424871316170, 5.15046571234168559435662086201, 6.04086982153802643747756652608, 7.12185528842530071708842988684, 7.71568923818544979862245115445, 8.497333007816474565722379669798, 9.867217429304536041151472084286, 11.14489306168793710514291423851

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