↑

Properties

Label 2-280-56.19-c1-0-22
Degree $2$
Conductor $280$
Sign $0.553 + 0.832i$
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  + (1.22 − 0.707i)2-s + (−0.275 + 0.158i)3-s + (0.999 − 1.73i)4-s + (0.5 − 0.866i)5-s + (−0.224 + 0.389i)6-s + (2.5 + 0.866i)7-s − 2.82i·8-s + (−1.44 + 2.51i)9-s − 1.41i·10-s + (−2.44 − 4.24i)11-s + 0.635i·12-s + 4.44·13-s + (3.67 − 0.707i)14-s + 0.317i·15-s + (−2.00 − 3.46i)16-s + (−4.22 + 2.43i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (−0.158 + 0.0917i)3-s + (0.499 − 0.866i)4-s + (0.223 − 0.387i)5-s + (−0.0917 + 0.158i)6-s + (0.944 + 0.327i)7-s − 0.999i·8-s + (−0.483 + 0.836i)9-s − 0.447i·10-s + (−0.738 − 1.27i)11-s + 0.183i·12-s + 1.23·13-s + (0.981 − 0.188i)14-s + 0.0820i·15-s + (−0.500 − 0.866i)16-s + (−1.02 + 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83360 - 0.982526i\)
\(L(\frac12)\) \(\approx\) \(1.83360 - 0.982526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good3 \( 1 + (0.275 - 0.158i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.44T + 13T^{2} \)
17 \( 1 + (4.22 - 2.43i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.67 - 2.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.94 + 2.28i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (-0.775 - 1.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 - 1.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.02iT - 41T^{2} \)
43 \( 1 + 9.44T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.77 + 1.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.12 + 1.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.174 - 0.301i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.17 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (9.67 - 5.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 - 3.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.87iT - 83T^{2} \)
89 \( 1 + (-9.39 - 5.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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

−11.46188015665774775086300103947, −11.10021609639586598897497845258, −10.27345101727129873483383992218, −8.694722485074362798089325082051, −8.070569266344549656308283504342, −6.22363257781224593399191038692, −5.49838008688405506348044011872, −4.60598100220317454596140416955, −3.12186617650362319691617646812, −1.63655223430770615177351639167, 2.25465609838410633543590467626, 3.79905384661436785738853675775, 4.91939563703591071652935929127, 5.97536995224891624199793017354, 6.99699499171948781973618803398, 7.82681331599712041657470485813, 8.962433653652879724439308214426, 10.33022673414576073852857313490, 11.51687414714500097939744470315, 11.80243761036085459606748741853

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