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Properties

Label 2-1120-56.19-c1-0-7
Degree $2$
Conductor $1120$
Sign $0.231 - 0.972i$
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  + (−2.26 + 1.30i)3-s + (−0.5 + 0.866i)5-s + (−1.72 − 2.00i)7-s + (1.92 − 3.33i)9-s + (−0.530 − 0.919i)11-s + 0.831·13-s − 2.61i·15-s + (4.14 − 2.39i)17-s + (2.03 + 1.17i)19-s + (6.53 + 2.28i)21-s + (−1.32 − 0.763i)23-s + (−0.499 − 0.866i)25-s + 2.23i·27-s + 3.07i·29-s + (4.78 + 8.28i)31-s + ⋯
L(s)  = 1  + (−1.30 + 0.755i)3-s + (−0.223 + 0.387i)5-s + (−0.652 − 0.757i)7-s + (0.642 − 1.11i)9-s + (−0.160 − 0.277i)11-s + 0.230·13-s − 0.675i·15-s + (1.00 − 0.580i)17-s + (0.467 + 0.269i)19-s + (1.42 + 0.498i)21-s + (−0.275 − 0.159i)23-s + (−0.0999 − 0.173i)25-s + 0.429i·27-s + 0.571i·29-s + (0.859 + 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.231 - 0.972i$
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: \((2,\ 1120,\ (\ :1/2),\ 0.231 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7350610343\)
\(L(\frac12)\) \(\approx\) \(0.7350610343\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 2.00i)T \)
good3 \( 1 + (2.26 - 1.30i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.530 + 0.919i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.831T + 13T^{2} \)
17 \( 1 + (-4.14 + 2.39i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.03 - 1.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.32 + 0.763i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.07iT - 29T^{2} \)
31 \( 1 + (-4.78 - 8.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.0 + 5.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.76iT - 41T^{2} \)
43 \( 1 - 4.99T + 43T^{2} \)
47 \( 1 + (-1.75 + 3.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.61 - 3.81i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.31 + 1.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.36 - 12.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.99iT - 71T^{2} \)
73 \( 1 + (7.06 - 4.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.17 + 4.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + (-10.1 - 5.88i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.01iT - 97T^{2} \)
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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

−10.17635403594098388240153634249, −9.600015905512273596011425566555, −8.363174427345058945134583925951, −7.25551012620543161768011165215, −6.59855168419941250030748222336, −5.68042087640051229853007925820, −4.95964617995037780119556700571, −3.88939166150170272535467155755, −3.11568868688768447303501685703, −0.901469138027036055980572849430, 0.53871118259545005966947645984, 1.91445660409884503663837745989, 3.38583712425350648086488211918, 4.71126540874064623435478592487, 5.70575624014571419248682513147, 6.04601688396886383228242058909, 7.06029165963693665884116614260, 7.84333211019822525088119338018, 8.800766469131726761181449752888, 9.798033903457013258816384416601

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