↑

Properties

Label 2-280-280.139-c1-0-25
Degree $2$
Conductor $280$
Sign $0.673 - 0.739i$
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  + (1.27 + 0.603i)2-s + 1.64·3-s + (1.27 + 1.54i)4-s + (−1.78 + 1.34i)5-s + (2.10 + 0.995i)6-s + (0.157 − 2.64i)7-s + (0.693 + 2.74i)8-s − 0.279·9-s + (−3.09 + 0.635i)10-s + 4.28·11-s + (2.09 + 2.54i)12-s + 0.653i·13-s + (1.79 − 3.28i)14-s + (−2.95 + 2.21i)15-s + (−0.768 + 3.92i)16-s − 2.74·17-s + ⋯
L(s)  = 1  + (0.904 + 0.426i)2-s + 0.952·3-s + (0.635 + 0.772i)4-s + (−0.800 + 0.599i)5-s + (0.861 + 0.406i)6-s + (0.0594 − 0.998i)7-s + (0.245 + 0.969i)8-s − 0.0930·9-s + (−0.979 + 0.200i)10-s + 1.29·11-s + (0.605 + 0.735i)12-s + 0.181i·13-s + (0.479 − 0.877i)14-s + (−0.761 + 0.571i)15-s + (−0.192 + 0.981i)16-s − 0.666·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25788 + 0.997011i\)
\(L(\frac12)\) \(\approx\) \(2.25788 + 0.997011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 - 0.603i)T \)
5 \( 1 + (1.78 - 1.34i)T \)
7 \( 1 + (-0.157 + 2.64i)T \)
good3 \( 1 - 1.64T + 3T^{2} \)
11 \( 1 - 4.28T + 11T^{2} \)
13 \( 1 - 0.653iT - 13T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 + 3.03iT - 19T^{2} \)
23 \( 1 + 7.21T + 23T^{2} \)
29 \( 1 + 7.72iT - 29T^{2} \)
31 \( 1 - 5.52T + 31T^{2} \)
37 \( 1 + 1.78T + 37T^{2} \)
41 \( 1 - 3.66iT - 41T^{2} \)
43 \( 1 + 6.86iT - 43T^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 - 7.10T + 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 0.822iT - 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + 5.19T + 73T^{2} \)
79 \( 1 + 1.45iT - 79T^{2} \)
83 \( 1 - 4.62T + 83T^{2} \)
89 \( 1 + 14.0iT - 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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

−11.81076639768527943659569970677, −11.48477520335487000728782607296, −10.19088959405202423310542784304, −8.773168457388801264109235614781, −7.930182992217664384195692379332, −7.07637517832722674619555094495, −6.23821829110686353755956767162, −4.25510983216321095868081846545, −3.83882994323230635266967957990, −2.51232241752792411518260785692, 1.88889099229285893495126945191, 3.30090885067006506972173550379, 4.18487756658978957702101720257, 5.47091177930877975619290527372, 6.63243555619512364930835631083, 8.083974222990988795661291760671, 8.837589134681369902019277731625, 9.723050953336306153094264195753, 11.19758143438395377003762089955, 11.99023663146269829656232730630

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