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Properties

Label 2-1120-56.19-c1-0-13
Degree $2$
Conductor $1120$
Sign $0.712 + 0.701i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.94 + 1.12i)3-s + (−0.5 + 0.866i)5-s + (−2.47 + 0.947i)7-s + (1.02 − 1.78i)9-s + (−0.656 − 1.13i)11-s − 3.02·13-s − 2.24i·15-s + (−0.313 + 0.181i)17-s + (−3.16 − 1.82i)19-s + (3.74 − 4.62i)21-s + (5.74 + 3.31i)23-s + (−0.499 − 0.866i)25-s − 2.12i·27-s + 3.35i·29-s + (−3.37 − 5.84i)31-s + ⋯
L(s)  = 1  + (−1.12 + 0.649i)3-s + (−0.223 + 0.387i)5-s + (−0.933 + 0.358i)7-s + (0.342 − 0.593i)9-s + (−0.198 − 0.342i)11-s − 0.838·13-s − 0.580i·15-s + (−0.0760 + 0.0439i)17-s + (−0.725 − 0.419i)19-s + (0.817 − 1.00i)21-s + (1.19 + 0.691i)23-s + (−0.0999 − 0.173i)25-s − 0.408i·27-s + 0.622i·29-s + (−0.606 − 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.712 + 0.701i$
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: \((2,\ 1120,\ (\ :1/2),\ 0.712 + 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3999314907\)
\(L(\frac12)\) \(\approx\) \(0.3999314907\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.47 - 0.947i)T \)
good3 \( 1 + (1.94 - 1.12i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.656 + 1.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.02T + 13T^{2} \)
17 \( 1 + (0.313 - 0.181i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.16 + 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.74 - 3.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.35iT - 29T^{2} \)
31 \( 1 + (3.37 + 5.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.89 - 2.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.07iT - 41T^{2} \)
43 \( 1 - 5.01T + 43T^{2} \)
47 \( 1 + (6.23 - 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.39 - 1.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.5 + 6.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.04 + 8.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.897 - 1.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (-7.83 + 4.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.89 - 4.55i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 + (1.99 + 1.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.74iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.700908393306451584459290971530, −9.295833474628650742979215464960, −8.033149249124567576176630860657, −6.99367788777577476453913826686, −6.27829511361952585975553651568, −5.45821589028570455821152209984, −4.68252887926630055910650754102, −3.56247599914326461990302940692, −2.50992788853149540890138729205, −0.26836008373758318239501169777, 0.898989287204528934229790837249, 2.49419971035009051874307415606, 3.88784408871955766377549702806, 4.95054707813022269635654405942, 5.72306321134078271814784403017, 6.79965539800781039998707502546, 7.02997249629507238792086574941, 8.187058814074414587563355913200, 9.196921938958266727709992435733, 10.05695391103837283386814781081

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