Properties

Label 2-280-56.3-c1-0-17
Degree $2$
Conductor $280$
Sign $0.947 - 0.319i$
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  + (1.41 − 0.0753i)2-s + (0.784 + 0.452i)3-s + (1.98 − 0.212i)4-s + (0.5 + 0.866i)5-s + (1.14 + 0.580i)6-s + (−1.23 + 2.34i)7-s + (2.79 − 0.450i)8-s + (−1.08 − 1.88i)9-s + (0.771 + 1.18i)10-s + (0.620 − 1.07i)11-s + (1.65 + 0.733i)12-s − 4.31·13-s + (−1.56 + 3.39i)14-s + 0.905i·15-s + (3.90 − 0.846i)16-s + (−1.49 − 0.862i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0532i)2-s + (0.452 + 0.261i)3-s + (0.994 − 0.106i)4-s + (0.223 + 0.387i)5-s + (0.466 + 0.237i)6-s + (−0.466 + 0.884i)7-s + (0.987 − 0.159i)8-s + (−0.363 − 0.629i)9-s + (0.243 + 0.374i)10-s + (0.187 − 0.324i)11-s + (0.478 + 0.211i)12-s − 1.19·13-s + (−0.418 + 0.908i)14-s + 0.233i·15-s + (0.977 − 0.211i)16-s + (−0.362 − 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.947 - 0.319i$
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: \((2,\ 280,\ (\ :1/2),\ 0.947 - 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46366 + 0.404167i\)
\(L(\frac12)\) \(\approx\) \(2.46366 + 0.404167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0753i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.23 - 2.34i)T \)
good3 \( 1 + (-0.784 - 0.452i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.620 + 1.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.31T + 13T^{2} \)
17 \( 1 + (1.49 + 0.862i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.12 + 1.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.393 + 0.226i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.69iT - 29T^{2} \)
31 \( 1 + (0.133 - 0.231i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.24 - 2.45i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 - 1.73T + 43T^{2} \)
47 \( 1 + (-5.37 - 9.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.50 + 4.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.83 + 2.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.462 + 0.801i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.465 + 0.807i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.36iT - 71T^{2} \)
73 \( 1 + (6.21 + 3.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.56 + 5.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.49iT - 83T^{2} \)
89 \( 1 + (6.91 - 3.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.92iT - 97T^{2} \)
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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

−11.96373586866155454423937445960, −11.31441967577906632787801997227, −9.930317055290528307782093215001, −9.291849983272503811608426219091, −7.924689894938614739694050394732, −6.64963803244924956741492519538, −5.89507850057336644763250051258, −4.67210879644423896596677976292, −3.23566082122513083448521409715, −2.49253254303321471299563435316, 1.97759869973888678283079649610, 3.32977760408508202782300750839, 4.61043586124054561231472862710, 5.57794183150376351173858903696, 7.03492990267964074141697737533, 7.51463457800072958990621248958, 8.878723899603816380638508670344, 10.14963989346648396917588296821, 10.91262361723822090741691453908, 12.22660472581370202373839542422

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