↑

Properties

Label 2-280-280.13-c1-0-5
Degree $2$
Conductor $280$
Sign $-0.117 - 0.993i$
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 + i)2-s + (−0.359 − 0.359i)3-s − 2i·4-s + (1.39 + 1.75i)5-s + 0.718·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s − 2.74i·9-s + (−3.14 − 0.359i)10-s + (−0.718 + 0.718i)12-s + (4.48 + 4.48i)13-s − 3.74i·14-s + (0.129 − 1.12i)15-s − 4·16-s + (2.74 + 2.74i)18-s + 7.62i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.207 − 0.207i)3-s − i·4-s + (0.622 + 0.782i)5-s + 0.293·6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s − 0.913i·9-s + (−0.993 − 0.113i)10-s + (−0.207 + 0.207i)12-s + (1.24 + 1.24i)13-s − 0.999i·14-s + (0.0333 − 0.291i)15-s − 16-s + (0.646 + 0.646i)18-s + 1.75i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.117 - 0.993i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.117 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561960 + 0.632468i\)
\(L(\frac12)\) \(\approx\) \(0.561960 + 0.632468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 + (-1.39 - 1.75i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (0.359 + 0.359i)T + 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.48 - 4.48i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 7.62iT - 19T^{2} \)
23 \( 1 + (-0.741 - 0.741i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 15.3iT - 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 7.22T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 15.7iT - 79T^{2} \)
83 \( 1 + (5.83 + 5.83i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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

−11.93064109992583062004915563595, −11.03621187734566477737628303996, −9.908010940668990640838408590411, −9.327445282296664241191161818758, −8.375361202849063989251412254822, −6.91867954036227487615623472397, −6.30142861691398043696474330590, −5.69878819543725489885940369249, −3.62214230631845193078223354374, −1.76746752373120844314252423008, 0.889490430849294701351055520748, 2.70476761719547101851873728900, 4.15250423914634092087162403678, 5.40611655372319100474273376780, 6.81956648180663733038786068730, 8.051555979416968590156414614600, 8.880816417391491479158825675377, 9.863688242302609941487565187744, 10.60535894330241018609411506666, 11.25628122131299229150101245714

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