↑

Properties

Label 2-280-35.13-c1-0-11
Degree $2$
Conductor $280$
Sign $-0.151 + 0.988i$
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  + (2.16 − 2.16i)3-s + (−1.91 − 1.14i)5-s + (−2.61 + 0.409i)7-s − 6.36i·9-s + 0.796·11-s + (3.25 − 3.25i)13-s + (−6.63 + 1.66i)15-s + (2.52 + 2.52i)17-s + 2.29·19-s + (−4.76 + 6.54i)21-s + (−2.08 − 2.08i)23-s + (2.35 + 4.40i)25-s + (−7.27 − 7.27i)27-s + 10.0i·29-s + 3.63i·31-s + ⋯
L(s)  = 1  + (1.24 − 1.24i)3-s + (−0.857 − 0.513i)5-s + (−0.987 + 0.154i)7-s − 2.12i·9-s + 0.240·11-s + (0.901 − 0.901i)13-s + (−1.71 + 0.429i)15-s + (0.611 + 0.611i)17-s + 0.527·19-s + (−1.04 + 1.42i)21-s + (−0.434 − 0.434i)23-s + (0.471 + 0.881i)25-s + (−1.39 − 1.39i)27-s + 1.87i·29-s + 0.652i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.151 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996431 - 1.16100i\)
\(L(\frac12)\) \(\approx\) \(0.996431 - 1.16100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 + 1.14i)T \)
7 \( 1 + (2.61 - 0.409i)T \)
good3 \( 1 + (-2.16 + 2.16i)T - 3iT^{2} \)
11 \( 1 - 0.796T + 11T^{2} \)
13 \( 1 + (-3.25 + 3.25i)T - 13iT^{2} \)
17 \( 1 + (-2.52 - 2.52i)T + 17iT^{2} \)
19 \( 1 - 2.29T + 19T^{2} \)
23 \( 1 + (2.08 + 2.08i)T + 23iT^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 - 3.63iT - 31T^{2} \)
37 \( 1 + (-7.30 + 7.30i)T - 37iT^{2} \)
41 \( 1 + 2.81iT - 41T^{2} \)
43 \( 1 + (-2.33 - 2.33i)T + 43iT^{2} \)
47 \( 1 + (4.09 + 4.09i)T + 47iT^{2} \)
53 \( 1 + (-6.50 - 6.50i)T + 53iT^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 3.35iT - 61T^{2} \)
67 \( 1 + (2.49 - 2.49i)T - 67iT^{2} \)
71 \( 1 + 2.93T + 71T^{2} \)
73 \( 1 + (-4.93 + 4.93i)T - 73iT^{2} \)
79 \( 1 - 6.53iT - 79T^{2} \)
83 \( 1 + (-1.14 + 1.14i)T - 83iT^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + (-4.30 - 4.30i)T + 97iT^{2} \)
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

−12.13408966259213263140997216381, −10.66614218935954711327851284588, −9.261457409191515699774110648462, −8.596543872596022359425146545665, −7.80173546490119748870147966264, −6.96429597717396001548491572890, −5.81311004039665069099342873878, −3.75676985283322076168339288117, −3.00963776078436051367473370894, −1.14735913220121699054176173695, 2.80239403386180202442684453034, 3.68649044105491029658751603496, 4.39335349009538965446262952639, 6.21677439864828734020703336102, 7.53886388516474516260038410489, 8.362253099636227788311096185672, 9.542707623133014283601638734069, 9.838456268825641992296219634526, 11.08299833112579566756812754262, 11.86945723065735153496855680775

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