Properties

Label 2-280-56.19-c1-0-7
Degree $2$
Conductor $280$
Sign $0.999 + 0.00549i$
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  + (−0.771 − 1.18i)2-s + (0.784 − 0.452i)3-s + (−0.810 + 1.82i)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 + (1.41 − 0.0753i)10-s + (0.620 + 1.07i)11-s + (0.192 + 1.80i)12-s + 4.31·13-s + (1.82 − 3.26i)14-s + 0.905i·15-s + (−2.68 − 2.96i)16-s + (−1.49 + 0.862i)17-s + ⋯
L(s)  = 1  + (−0.545 − 0.838i)2-s + (0.452 − 0.261i)3-s + (−0.405 + 0.914i)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 + (0.446 − 0.0238i)10-s + (0.187 + 0.324i)11-s + (0.0556 + 0.520i)12-s + 1.19·13-s + (0.487 − 0.873i)14-s + 0.233i·15-s + (−0.671 − 0.740i)16-s + (−0.362 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.00549i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.999 + 0.00549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08608 - 0.00298479i\)
\(L(\frac12)\) \(\approx\) \(1.08608 - 0.00298479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.771 + 1.18i)T \)
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

−11.58245971706297193853702912598, −11.11828728864670947507583772351, −10.01178921566247376980317975454, −8.856061907569864070246592054704, −8.285899940160855644497657490596, −7.39112444123939188208053716471, −5.83522904551977328729802866241, −4.28278622540693847974427206241, −2.93240648056426129836729401949, −1.83863141936335647326093042342, 1.07525690111500490256225010302, 3.59914166830849374853372665255, 4.72582471170412469709278941025, 6.03783520970281759943514268013, 7.07584888639994938567098153631, 8.186089188498722065398498625779, 8.801579897243080915064997607370, 9.669369221569933247366444299205, 10.81098207189107369633038185976, 11.55235616130825048765080862382

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