↑

Properties

Label 2-1470-7.2-c1-0-14
Degree $2$
Conductor $1470$
Sign $0.827 + 0.561i$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−1.12 − 1.94i)11-s + (0.499 − 0.866i)12-s + 5.65·13-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−1.29 − 2.23i)17-s + (0.499 + 0.866i)18-s + (3.41 − 5.91i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.338 − 0.585i)11-s + (0.144 − 0.249i)12-s + 1.56·13-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.313 − 0.543i)17-s + (0.117 + 0.204i)18-s + (0.783 − 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.184307680\)
\(L(\frac12)\) \(\approx\) \(2.184307680\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (1.12 + 1.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + (1.29 + 2.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.41 + 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.58 - 2.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.58T + 29T^{2} \)
31 \( 1 + (-5.12 - 8.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.121 + 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 9.07T + 43T^{2} \)
47 \( 1 + (-1.12 + 1.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.171 + 0.297i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.58 - 2.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.82 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.53 + 7.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.17T + 83T^{2} \)
89 \( 1 + (-0.414 + 0.717i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.48T + 97T^{2} \)
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

−9.413496341472916910480713185708, −8.834314108507481954087888841697, −8.008470576856028317638502452668, −6.92822969445523115940061267278, −5.99400913448138292456619438970, −5.10595233827270097886405493201, −4.16768490600048719273155204882, −3.27752304117845122469781591050, −2.62556243675224945807149190262, −0.971246752159704847088851967305, 1.12941374969799924950738779973, 2.52631863295612803962562540387, 3.82701888269499882794564797050, 4.38661367966302837001874434964, 5.83158028240392012653824663963, 6.08745915132605270591172557181, 7.27462062345296340142041362458, 7.979583074440070788240689621833, 8.470453277216158517905624174927, 9.354690939728712469110132356186

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