↑

Properties

Label 2-1470-35.13-c1-0-10
Degree $2$
Conductor $1470$
Sign $0.994 + 0.102i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−2.15 − 0.582i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−1.93 + 1.11i)10-s + 0.461·11-s + (0.707 + 0.707i)12-s + (−4.00 + 4.00i)13-s + (1.93 − 1.11i)15-s − 1.00·16-s + (1.15 + 1.15i)17-s + (−0.707 − 0.707i)18-s + 5.82·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (−0.965 − 0.260i)5-s + 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (−0.612 + 0.352i)10-s + 0.139·11-s + (0.204 + 0.204i)12-s + (−1.11 + 1.11i)13-s + (0.500 − 0.287i)15-s − 0.250·16-s + (0.281 + 0.281i)17-s + (−0.166 − 0.166i)18-s + 1.33·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.994 + 0.102i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.994 + 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451328848\)
\(L(\frac12)\) \(\approx\) \(1.451328848\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.15 + 0.582i)T \)
7 \( 1 \)
good11 \( 1 - 0.461T + 11T^{2} \)
13 \( 1 + (4.00 - 4.00i)T - 13iT^{2} \)
17 \( 1 + (-1.15 - 1.15i)T + 17iT^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 + (-3.11 - 3.11i)T + 23iT^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 - 0.0324iT - 31T^{2} \)
37 \( 1 + (-5.70 + 5.70i)T - 37iT^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (-4.75 - 4.75i)T + 43iT^{2} \)
47 \( 1 + (-6.39 - 6.39i)T + 47iT^{2} \)
53 \( 1 + (-1.94 - 1.94i)T + 53iT^{2} \)
59 \( 1 + 1.91T + 59T^{2} \)
61 \( 1 - 13.5iT - 61T^{2} \)
67 \( 1 + (2.69 - 2.69i)T - 67iT^{2} \)
71 \( 1 - 8.85T + 71T^{2} \)
73 \( 1 + (-2.86 + 2.86i)T - 73iT^{2} \)
79 \( 1 - 5.06iT - 79T^{2} \)
83 \( 1 + (-1.08 + 1.08i)T - 83iT^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (-2.51 - 2.51i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.429464341588288230559750393145, −9.088037598026670288451446627876, −7.64594600383618012394658365085, −7.19738661643470429132957327037, −5.96581217500921148539395097347, −5.11804426193402521793198721381, −4.33460010676672503815340287127, −3.69209456947523377137617553007, −2.51278320263658793256322806823, −0.926869631368609347642062534318, 0.72402158469151550921276787284, 2.72668735673197138779240328821, 3.44721868987305729793668760272, 4.77599873504715216070263934559, 5.21897585929490520167404965195, 6.34634018060984424594179668422, 7.19544908786844807850078659680, 7.64448476729998596894467069198, 8.354600188325696198854145178119, 9.490753480714506616481982453558

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