Properties

Label 2-280-56.3-c1-0-14
Degree $2$
Conductor $280$
Sign $0.613 - 0.790i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 0.311i)2-s + (2.75 + 1.58i)3-s + (1.80 − 0.858i)4-s + (0.5 + 0.866i)5-s + (−4.29 − 1.33i)6-s + (1.04 − 2.43i)7-s + (−2.22 + 1.74i)8-s + (3.55 + 6.15i)9-s + (−0.959 − 1.03i)10-s + (1.21 − 2.09i)11-s + (6.33 + 0.507i)12-s + 1.53·13-s + (−0.681 + 3.67i)14-s + 3.17i·15-s + (2.52 − 3.10i)16-s + (−6.58 − 3.79i)17-s + ⋯
L(s)  = 1  + (−0.975 + 0.220i)2-s + (1.58 + 0.917i)3-s + (0.903 − 0.429i)4-s + (0.223 + 0.387i)5-s + (−1.75 − 0.545i)6-s + (0.394 − 0.919i)7-s + (−0.786 + 0.617i)8-s + (1.18 + 2.05i)9-s + (−0.303 − 0.328i)10-s + (0.364 − 0.631i)11-s + (1.82 + 0.146i)12-s + 0.426·13-s + (−0.182 + 0.983i)14-s + 0.820i·15-s + (0.631 − 0.775i)16-s + (−1.59 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.613 - 0.790i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.613 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31619 + 0.644647i\)
\(L(\frac12)\) \(\approx\) \(1.31619 + 0.644647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.311i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.04 + 2.43i)T \)
good3 \( 1 + (-2.75 - 1.58i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.21 + 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.53T + 13T^{2} \)
17 \( 1 + (6.58 + 3.79i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.52 - 0.883i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.66 - 3.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.34iT - 29T^{2} \)
31 \( 1 + (-1.04 + 1.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.47 - 1.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.90iT - 41T^{2} \)
43 \( 1 + 1.39T + 43T^{2} \)
47 \( 1 + (-5.65 - 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.82 + 3.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.90 - 2.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.93 + 8.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.13 + 5.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.98iT - 71T^{2} \)
73 \( 1 + (2.73 + 1.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.75 + 1.01i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.288iT - 83T^{2} \)
89 \( 1 + (-12.0 + 6.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.249iT - 97T^{2} \)
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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

−11.39322910477920521933952924787, −10.74477969358709379292506563474, −9.847003143829998734129023844297, −9.175567856268473628186432561529, −8.277745180714150334807855087865, −7.57797314929257778761738178818, −6.37995254465740214523708821250, −4.52148928919604301702555592671, −3.31715751996844700629619070456, −2.01167745857422690291565473753, 1.77071775952797101663270655212, 2.37428408723785505198414341613, 3.98045378393568894590688740747, 6.25132047604590222755351436558, 7.11924639876268403237413432033, 8.372472871538759043267991464164, 8.635738296038420495984586757764, 9.352498649057883053874157163552, 10.55762769152795885067090683060, 11.97133053247921003708382820395

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