↑

Properties

Label 2-1120-56.19-c1-0-9
Degree $2$
Conductor $1120$
Sign $0.444 - 0.895i$
Analytic cond. $8.94324$
Root an. cond. $2.99052$
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  + (0.275 − 0.158i)3-s + (0.5 − 0.866i)5-s + (−2.5 − 0.866i)7-s + (−1.44 + 2.51i)9-s + (2.44 + 4.24i)11-s + 4.44·13-s − 0.317i·15-s + (−4.22 + 2.43i)17-s + (−3.67 − 2.12i)19-s + (−0.825 + 0.158i)21-s + (3.94 + 2.28i)23-s + (−0.499 − 0.866i)25-s + 1.87i·27-s + 7.24i·29-s + (−0.775 − 1.34i)31-s + ⋯
L(s)  = 1  + (0.158 − 0.0917i)3-s + (0.223 − 0.387i)5-s + (−0.944 − 0.327i)7-s + (−0.483 + 0.836i)9-s + (0.738 + 1.27i)11-s + 1.23·13-s − 0.0820i·15-s + (−1.02 + 0.591i)17-s + (−0.842 − 0.486i)19-s + (−0.180 + 0.0346i)21-s + (0.823 + 0.475i)23-s + (−0.0999 − 0.173i)25-s + 0.360i·27-s + 1.34i·29-s + (−0.139 − 0.241i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.396810067\)
\(L(\frac12)\) \(\approx\) \(1.396810067\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
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

−9.931775860118906867491253967526, −8.972785467592928948455715616398, −8.644253301749308788450762992601, −7.36323103761743557325852841216, −6.66861568558550408184956132059, −5.86291094574242785060513042654, −4.66950786921828162330013870541, −3.89865986107515988730131127957, −2.61774312295589630903114504206, −1.43758681576647154994897948180, 0.62998320127379467896886470893, 2.50439849398684380663236384634, 3.41606276787399664746869362089, 4.13482568298088106701251645968, 5.97808096796991156672489323744, 6.08559146362905356075743800825, 6.92143966469345978686218428065, 8.332352402881726558984583085463, 9.038949806092924196781922523637, 9.347287440547929679757511589642

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