Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.164 - 0.986i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.790 − 1.54i)3-s + (0.499 − 0.866i)4-s − 5-s + (1.45 + 0.939i)6-s + (−1.82 − 1.91i)7-s + 0.999i·8-s + (−1.75 + 2.43i)9-s + (0.866 − 0.5i)10-s + 3.07i·11-s + (−1.72 − 0.0862i)12-s + (−0.449 + 0.259i)13-s + (2.53 + 0.752i)14-s + (0.790 + 1.54i)15-s + (−0.5 − 0.866i)16-s + (−2.44 − 4.23i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.456 − 0.889i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.594 + 0.383i)6-s + (−0.688 − 0.725i)7-s + 0.353i·8-s + (−0.583 + 0.811i)9-s + (0.273 − 0.158i)10-s + 0.927i·11-s + (−0.499 − 0.0248i)12-s + (−0.124 + 0.0720i)13-s + (0.677 + 0.200i)14-s + (0.204 + 0.397i)15-s + (−0.125 − 0.216i)16-s + (−0.592 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.164 - 0.986i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (551, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.164 - 0.986i)\)
\(L(1)\)  \(\approx\)  \(0.311148 + 0.263477i\)
\(L(\frac12)\)  \(\approx\)  \(0.311148 + 0.263477i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.790 + 1.54i)T \)
5 \( 1 + T \)
7 \( 1 + (1.82 + 1.91i)T \)
good11 \( 1 - 3.07iT - 11T^{2} \)
13 \( 1 + (0.449 - 0.259i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.44 + 4.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.713 - 0.412i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.94iT - 23T^{2} \)
29 \( 1 + (-5.55 - 3.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.784 + 0.452i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.53 + 4.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.18 - 3.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.84 - 3.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.89 - 3.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.69 + 2.71i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.29 - 9.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.24 - 4.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.95 - 6.84i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 + (-5.78 + 3.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.73 + 2.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.32 - 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.771 + 1.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.93 - 1.11i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.80252795260899015269239752108, −9.845463884664174271544294415283, −9.075185300727031511750853384235, −7.79155478685769356265134490360, −7.26980460042501638473285164769, −6.70002402166422494166757700924, −5.56549298110731895861365363347, −4.41953084699913369564690715919, −2.80031181933149207167829855532, −1.22687392638480392922718558731, 0.31915890134172177719732803373, 2.64387293079988513155892125774, 3.62191194100845052867377740784, 4.69656223694475659865641688371, 6.02832133432987366834954750889, 6.60663996042678860283856259018, 8.250651073823039850035430333016, 8.719841687406315970229543897941, 9.576612600320857950673251365048, 10.54235192091980752587287195924

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