↑

Properties

Label 2-280-56.27-c1-0-2
Degree $2$
Conductor $280$
Sign $-0.709 + 0.705i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (−0.275 + 1.38i)2-s + 3.19i·3-s + (−1.84 − 0.764i)4-s + 5-s + (−4.43 − 0.881i)6-s + (−2.59 − 0.525i)7-s + (1.56 − 2.35i)8-s − 7.23·9-s + (−0.275 + 1.38i)10-s − 3.34·11-s + (2.44 − 5.91i)12-s + 3.90·13-s + (1.44 − 3.45i)14-s + 3.19i·15-s + (2.83 + 2.82i)16-s + 2.92i·17-s + ⋯
L(s)  = 1  + (−0.194 + 0.980i)2-s + 1.84i·3-s + (−0.924 − 0.382i)4-s + 0.447·5-s + (−1.81 − 0.359i)6-s + (−0.980 − 0.198i)7-s + (0.555 − 0.831i)8-s − 2.41·9-s + (−0.0871 + 0.438i)10-s − 1.00·11-s + (0.706 − 1.70i)12-s + 1.08·13-s + (0.385 − 0.922i)14-s + 0.826i·15-s + (0.707 + 0.706i)16-s + 0.710i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.709 + 0.705i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.709 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293358 - 0.711181i\)
\(L(\frac12)\) \(\approx\) \(0.293358 - 0.711181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.275 - 1.38i)T \)
5 \( 1 - T \)
7 \( 1 + (2.59 + 0.525i)T \)
good3 \( 1 - 3.19iT - 3T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
13 \( 1 - 3.90T + 13T^{2} \)
17 \( 1 - 2.92iT - 17T^{2} \)
19 \( 1 - 6.33iT - 19T^{2} \)
23 \( 1 - 3.44iT - 23T^{2} \)
29 \( 1 + 2.68iT - 29T^{2} \)
31 \( 1 + 2.52T + 31T^{2} \)
37 \( 1 + 4.70iT - 37T^{2} \)
41 \( 1 - 5.59iT - 41T^{2} \)
43 \( 1 - 8.62T + 43T^{2} \)
47 \( 1 + 0.506T + 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + 0.802iT - 59T^{2} \)
61 \( 1 + 7.97T + 61T^{2} \)
67 \( 1 - 6.37T + 67T^{2} \)
71 \( 1 - 1.11iT - 71T^{2} \)
73 \( 1 - 5.91iT - 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 - 4.29iT - 83T^{2} \)
89 \( 1 - 2.00iT - 89T^{2} \)
97 \( 1 + 6.06iT - 97T^{2} \)
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

−12.69305850001916145535999755877, −10.88416384147512182310474746887, −10.29323828224605743706261598690, −9.616243574781517363108704002388, −8.838324053729007945714856503727, −7.81102285727822377110605816565, −6.00784915362688519673664320998, −5.68692281575743796809513988869, −4.26927858709043131877977535389, −3.40762692700717015302216192792, 0.61740747125339861430811793903, 2.25667426603866610563988639939, 3.08738565641852910157477153139, 5.28571005562175545496958622667, 6.41735376618122429902523102040, 7.38078513919822555527264116218, 8.517694366766836029112600976274, 9.218085573017302002413786321662, 10.58278513703218287877187885424, 11.41374534506783017218207751297

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