↑

Properties

Label 2-1890-63.59-c1-0-15
Degree $2$
Conductor $1890$
Sign $0.985 - 0.171i$
Analytic cond. $15.0917$
Root an. cond. $3.88480$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 5-s + (2.13 + 1.55i)7-s − 0.999i·8-s + (−0.866 − 0.5i)10-s + 0.450i·11-s + (4.26 + 2.46i)13-s + (−1.07 − 2.41i)14-s + (−0.5 + 0.866i)16-s + (3.93 − 6.81i)17-s + (−4.75 + 2.74i)19-s + (0.499 + 0.866i)20-s + (0.225 − 0.390i)22-s + 4.86i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.807 + 0.589i)7-s − 0.353i·8-s + (−0.273 − 0.158i)10-s + 0.135i·11-s + (1.18 + 0.682i)13-s + (−0.286 − 0.646i)14-s + (−0.125 + 0.216i)16-s + (0.953 − 1.65i)17-s + (−1.08 + 0.629i)19-s + (0.111 + 0.193i)20-s + (0.0480 − 0.0831i)22-s + 1.01i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.985 - 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.675578526\)
\(L(\frac12)\) \(\approx\) \(1.675578526\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (-2.13 - 1.55i)T \)
good11 \( 1 - 0.450iT - 11T^{2} \)
13 \( 1 + (-4.26 - 2.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.93 + 6.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.75 - 2.74i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.86iT - 23T^{2} \)
29 \( 1 + (-8.05 + 4.65i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.497 + 0.287i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.721 + 1.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.956 - 1.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.459 + 0.795i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.71 - 6.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.30 + 4.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.43 + 7.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.54 - 4.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.88iT - 71T^{2} \)
73 \( 1 + (-4.91 - 2.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.00 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.76 - 8.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.98 - 3.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.69 - 5.01i)T + (48.5 - 84.0i)T^{2} \)
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

−9.400356406345030012668714870587, −8.385554111238674931966404785951, −8.036411311487525811258154315649, −6.92024707989315519870995684522, −6.11937428715480457872000515917, −5.22770246809852479381796873660, −4.26763155469454496843614018823, −3.10221585258722724298205337212, −2.08071923830633665129997982311, −1.16326061505331809881403585413, 0.932305389582447665211941391454, 1.85410043962964267855598222206, 3.25639970732000035245007764605, 4.35877037414708210641983750748, 5.29000932634488198676711119688, 6.24306220636309300896083205163, 6.69959714381717539183548030172, 7.966297628336120840014613441177, 8.331201931878666885375811754609, 8.919745830537606594859285428801

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