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Properties

Label 2-1120-56.3-c1-0-17
Degree $2$
Conductor $1120$
Sign $0.937 + 0.347i$
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  + (0.502 + 0.290i)3-s + (−0.5 − 0.866i)5-s + (2.63 + 0.253i)7-s + (−1.33 − 2.30i)9-s + (−0.428 + 0.742i)11-s + 2.26·13-s − 0.580i·15-s + (6.65 + 3.84i)17-s + (−5.17 + 2.98i)19-s + (1.25 + 0.891i)21-s + (3.17 − 1.83i)23-s + (−0.499 + 0.866i)25-s − 3.28i·27-s − 7.76i·29-s + (4.53 − 7.86i)31-s + ⋯
L(s)  = 1  + (0.290 + 0.167i)3-s + (−0.223 − 0.387i)5-s + (0.995 + 0.0956i)7-s + (−0.443 − 0.768i)9-s + (−0.129 + 0.223i)11-s + 0.627·13-s − 0.149i·15-s + (1.61 + 0.931i)17-s + (−1.18 + 0.684i)19-s + (0.272 + 0.194i)21-s + (0.661 − 0.381i)23-s + (−0.0999 + 0.173i)25-s − 0.632i·27-s − 1.44i·29-s + (0.815 − 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.937 + 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.942375459\)
\(L(\frac12)\) \(\approx\) \(1.942375459\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.253i)T \)
good3 \( 1 + (-0.502 - 0.290i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.428 - 0.742i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 + (-6.65 - 3.84i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.17 - 2.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.17 + 1.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.76iT - 29T^{2} \)
31 \( 1 + (-4.53 + 7.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.77 + 2.18i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.780iT - 41T^{2} \)
43 \( 1 + 7.36T + 43T^{2} \)
47 \( 1 + (-0.206 - 0.358i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.0 - 6.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.74 - 4.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.51 + 7.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.69iT - 71T^{2} \)
73 \( 1 + (9.52 + 5.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.53 - 2.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.58iT - 83T^{2} \)
89 \( 1 + (-5.85 + 3.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.09iT - 97T^{2} \)
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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

−9.829774474108776278553252148746, −8.706346864287092324078204619389, −8.290256608234910543698544198381, −7.60117023215471093552478916471, −6.19324632893947150068575137721, −5.64815050072023154557930594053, −4.35439777961771601061709480296, −3.76594498241059389306571386015, −2.38161094113731701945027160802, −1.02383988996871964224580504217, 1.26846323996553742896669156859, 2.61104536246756205951351322475, 3.49598116213405918511793028347, 4.89003572517513642350821587350, 5.38687377650406054558009945799, 6.74504997655257363461074935159, 7.44838671486783543223128971550, 8.377420749879847051119331189522, 8.673046564805588546574058719631, 10.02962530658440357234218990975

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