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Properties

Label 2-280-56.27-c1-0-12
Degree $2$
Conductor $280$
Sign $0.400 - 0.916i$
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  + (−0.470 + 1.33i)2-s + 0.528i·3-s + (−1.55 − 1.25i)4-s + 5-s + (−0.704 − 0.248i)6-s + (2.17 − 1.50i)7-s + (2.40 − 1.48i)8-s + 2.72·9-s + (−0.470 + 1.33i)10-s + 3.04·11-s + (0.663 − 0.822i)12-s − 4.75·13-s + (0.985 + 3.60i)14-s + 0.528i·15-s + (0.844 + 3.90i)16-s + 6.78i·17-s + ⋯
L(s)  = 1  + (−0.333 + 0.942i)2-s + 0.304i·3-s + (−0.778 − 0.628i)4-s + 0.447·5-s + (−0.287 − 0.101i)6-s + (0.821 − 0.569i)7-s + (0.851 − 0.524i)8-s + 0.907·9-s + (−0.148 + 0.421i)10-s + 0.918·11-s + (0.191 − 0.237i)12-s − 1.31·13-s + (0.263 + 0.964i)14-s + 0.136i·15-s + (0.211 + 0.977i)16-s + 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.400 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03444 + 0.676569i\)
\(L(\frac12)\) \(\approx\) \(1.03444 + 0.676569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.470 - 1.33i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.17 + 1.50i)T \)
good3 \( 1 - 0.528iT - 3T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 + 4.75T + 13T^{2} \)
17 \( 1 - 6.78iT - 17T^{2} \)
19 \( 1 + 0.584iT - 19T^{2} \)
23 \( 1 + 5.80iT - 23T^{2} \)
29 \( 1 + 0.185iT - 29T^{2} \)
31 \( 1 - 3.12T + 31T^{2} \)
37 \( 1 - 7.04iT - 37T^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 + 0.535iT - 53T^{2} \)
59 \( 1 + 1.52iT - 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + 9.13T + 67T^{2} \)
71 \( 1 + 9.68iT - 71T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 + 2.81iT - 79T^{2} \)
83 \( 1 + 13.4iT - 83T^{2} \)
89 \( 1 + 3.10iT - 89T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−12.14228496337508304518809713701, −10.57610788672612320979884196836, −10.15161970469342167888532674902, −9.114336017085564269223816964964, −8.115016154028922498660003756805, −7.12567886426798079024758108177, −6.25987937149316841481202695136, −4.83895923451092101092800841944, −4.17625232983370484405802455462, −1.54886346216681532228131749993, 1.45396513664675929370458057608, 2.64005542417613593550426469598, 4.36083819002557442810050634637, 5.30923863089597553140442344181, 7.07284570927092457009192814790, 7.85692849158345478356167681084, 9.320785008306851606929961935067, 9.555047838689482103242439138184, 10.81579084857764221444565586736, 11.96410451773751381454220535350

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