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Properties

Label 2-280-56.53-c1-0-30
Degree $2$
Conductor $280$
Sign $-0.999 + 0.00829i$
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  + (0.366 − 1.36i)2-s + (−1.73 − i)4-s + (−0.866 + 0.5i)5-s + (−2 − 1.73i)7-s + (−2 + 1.99i)8-s + (−1.5 − 2.59i)9-s + (0.366 + 1.36i)10-s + (−2.59 − 1.5i)11-s + i·13-s + (−3.09 + 2.09i)14-s + (1.99 + 3.46i)16-s + (3 − 5.19i)17-s + (−4.09 + 1.09i)18-s + (0.866 − 0.5i)19-s + 1.99·20-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.387 + 0.223i)5-s + (−0.755 − 0.654i)7-s + (−0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + (0.115 + 0.431i)10-s + (−0.783 − 0.452i)11-s + 0.277i·13-s + (−0.827 + 0.560i)14-s + (0.499 + 0.866i)16-s + (0.727 − 1.26i)17-s + (−0.965 + 0.258i)18-s + (0.198 − 0.114i)19-s + 0.447·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.00829i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.999 + 0.00829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00313726 - 0.756092i\)
\(L(\frac12)\) \(\approx\) \(0.00313726 - 0.756092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2 + 1.73i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 10iT - 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.33 + 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.92 + 4i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.92 + 4i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−11.38751548409034681653828671560, −10.59329714607059554849923728690, −9.659528946047868032906079057069, −8.841748061865999384069237794138, −7.50347576087130950605157656754, −6.26412335998463421655581646913, −5.03663744623152158850903780533, −3.62769896681003836067096958125, −2.89697032698345654887697502194, −0.52058936100865036501944638053, 2.84288772818364279041285092514, 4.27078179893400949542686112594, 5.51753420017068419624067499828, 6.16107949411951853832092919261, 7.80762674230977037557289278781, 8.021949561591049977955255770035, 9.339705278241834864094654828327, 10.22088455450106311131594632898, 11.60915681860465492486559583255, 12.67713277381860371288277996482

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