↑

Properties

Label 2-840-105.89-c1-0-21
Degree $2$
Conductor $840$
Sign $0.683 + 0.729i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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  + (0.395 − 1.68i)3-s + (−2.06 + 0.850i)5-s + (0.514 + 2.59i)7-s + (−2.68 − 1.33i)9-s + (0.513 − 0.296i)11-s + 3.52·13-s + (0.616 + 3.82i)15-s + (5.41 − 3.12i)17-s + (−0.455 − 0.262i)19-s + (4.57 + 0.158i)21-s + (3.96 − 6.87i)23-s + (3.55 − 3.51i)25-s + (−3.31 + 4.00i)27-s − 2.35i·29-s + (4.32 − 2.49i)31-s + ⋯
L(s)  = 1  + (0.228 − 0.973i)3-s + (−0.924 + 0.380i)5-s + (0.194 + 0.980i)7-s + (−0.895 − 0.444i)9-s + (0.154 − 0.0893i)11-s + 0.978·13-s + (0.159 + 0.987i)15-s + (1.31 − 0.758i)17-s + (−0.104 − 0.0602i)19-s + (0.999 + 0.0344i)21-s + (0.827 − 1.43i)23-s + (0.710 − 0.703i)25-s + (−0.637 + 0.770i)27-s − 0.436i·29-s + (0.776 − 0.448i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.683 + 0.729i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.683 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37610 - 0.596707i\)
\(L(\frac12)\) \(\approx\) \(1.37610 - 0.596707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.395 + 1.68i)T \)
5 \( 1 + (2.06 - 0.850i)T \)
7 \( 1 + (-0.514 - 2.59i)T \)
good11 \( 1 + (-0.513 + 0.296i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.52T + 13T^{2} \)
17 \( 1 + (-5.41 + 3.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.455 + 0.262i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.96 + 6.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.35iT - 29T^{2} \)
31 \( 1 + (-4.32 + 2.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.71 - 5.03i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.64T + 41T^{2} \)
43 \( 1 - 3.39iT - 43T^{2} \)
47 \( 1 + (7.55 + 4.36i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.45 - 7.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.45 + 2.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.2 - 5.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.49 + 1.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.36iT - 71T^{2} \)
73 \( 1 + (6.28 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.276 - 0.479i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.27iT - 83T^{2} \)
89 \( 1 + (1.52 - 2.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.81T + 97T^{2} \)
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

−10.09523629659033311274846543959, −8.902864675212256187497886410991, −8.307277157348972997440439855123, −7.66892671985587359221593397256, −6.63078967355172095753862459408, −5.97775217355076803178635270036, −4.73730606591718990657786022533, −3.32460560780059246175263559630, −2.57864185932400795789846581887, −0.931942187139901441441631332491, 1.17392143419382049148739889641, 3.38435060310437112951341386611, 3.79390677436486275196509577938, 4.77316360247574629920869712708, 5.70495447393120712814667200970, 7.05622906801478505324649385879, 8.002021423408786593446518845857, 8.501429030353847020466177338116, 9.559735284456003763557913428871, 10.29516752534488553130032192569

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