↑

Properties

Label 2-280-35.27-c1-0-2
Degree $2$
Conductor $280$
Sign $0.163 - 0.986i$
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  + (1.03 + 1.03i)3-s + (−1.21 + 1.87i)5-s + (1.58 + 2.11i)7-s − 0.841i·9-s − 2.34·11-s + (1.96 + 1.96i)13-s + (−3.21 + 0.679i)15-s + (−5.15 + 5.15i)17-s + 3.74·19-s + (−0.547 + 3.84i)21-s + (6.08 − 6.08i)23-s + (−2.02 − 4.57i)25-s + (3.99 − 3.99i)27-s + 5.89i·29-s − 1.56i·31-s + ⋯
L(s)  = 1  + (0.599 + 0.599i)3-s + (−0.545 + 0.838i)5-s + (0.600 + 0.799i)7-s − 0.280i·9-s − 0.707·11-s + (0.544 + 0.544i)13-s + (−0.829 + 0.175i)15-s + (−1.25 + 1.25i)17-s + 0.859·19-s + (−0.119 + 0.839i)21-s + (1.26 − 1.26i)23-s + (−0.404 − 0.914i)25-s + (0.768 − 0.768i)27-s + 1.09i·29-s − 0.281i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08940 + 0.923347i\)
\(L(\frac12)\) \(\approx\) \(1.08940 + 0.923347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.21 - 1.87i)T \)
7 \( 1 + (-1.58 - 2.11i)T \)
good3 \( 1 + (-1.03 - 1.03i)T + 3iT^{2} \)
11 \( 1 + 2.34T + 11T^{2} \)
13 \( 1 + (-1.96 - 1.96i)T + 13iT^{2} \)
17 \( 1 + (5.15 - 5.15i)T - 17iT^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 + (-6.08 + 6.08i)T - 23iT^{2} \)
29 \( 1 - 5.89iT - 29T^{2} \)
31 \( 1 + 1.56iT - 31T^{2} \)
37 \( 1 + (-1.53 - 1.53i)T + 37iT^{2} \)
41 \( 1 + 9.51iT - 41T^{2} \)
43 \( 1 + (-1.86 + 1.86i)T - 43iT^{2} \)
47 \( 1 + (-4.59 + 4.59i)T - 47iT^{2} \)
53 \( 1 + (-3.88 + 3.88i)T - 53iT^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 + 2.00iT - 61T^{2} \)
67 \( 1 + (-2.69 - 2.69i)T + 67iT^{2} \)
71 \( 1 + 0.392T + 71T^{2} \)
73 \( 1 + (7.08 + 7.08i)T + 73iT^{2} \)
79 \( 1 + 4.98iT - 79T^{2} \)
83 \( 1 + (9.36 + 9.36i)T + 83iT^{2} \)
89 \( 1 + 9.83T + 89T^{2} \)
97 \( 1 + (-11.8 + 11.8i)T - 97iT^{2} \)
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

−11.92869382413988206007313402625, −10.98293378013688574178541582643, −10.37051553242129520319096612560, −8.885684121267205773285188702686, −8.590151768641324142711773822597, −7.24090491974367396925050644320, −6.16121285876203241799486342545, −4.69762813692824843443275091604, −3.57492327187724603200178892606, −2.41710202413787496296632982285, 1.14651663882308992872207997871, 2.89311440286054397651391383380, 4.45480683680093081339306400637, 5.34673843858752142328424254498, 7.22776813652062655922340654705, 7.71376215994080109362392613040, 8.555975391615726213772106531906, 9.566655311973993374209984009003, 10.99584498890399020496926170443, 11.50362431663289334145126846853

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