Properties

Label 2-280-56.27-c1-0-7
Degree $2$
Conductor $280$
Sign $0.914 - 0.405i$
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  + (−1.38 − 0.303i)2-s + 1.34i·3-s + (1.81 + 0.837i)4-s − 5-s + (0.406 − 1.85i)6-s + (1.28 − 2.31i)7-s + (−2.25 − 1.70i)8-s + 1.20·9-s + (1.38 + 0.303i)10-s + 2.44·11-s + (−1.12 + 2.43i)12-s + 1.57·13-s + (−2.47 + 2.81i)14-s − 1.34i·15-s + (2.59 + 3.04i)16-s − 1.11i·17-s + ⋯
L(s)  = 1  + (−0.976 − 0.214i)2-s + 0.774i·3-s + (0.908 + 0.418i)4-s − 0.447·5-s + (0.166 − 0.756i)6-s + (0.483 − 0.875i)7-s + (−0.797 − 0.603i)8-s + 0.400·9-s + (0.436 + 0.0958i)10-s + 0.738·11-s + (−0.324 + 0.703i)12-s + 0.435·13-s + (−0.660 + 0.751i)14-s − 0.346i·15-s + (0.649 + 0.760i)16-s − 0.271i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.914 - 0.405i$
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: \((2,\ 280,\ (\ :1/2),\ 0.914 - 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881731 + 0.186826i\)
\(L(\frac12)\) \(\approx\) \(0.881731 + 0.186826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.303i)T \)
5 \( 1 + T \)
7 \( 1 + (-1.28 + 2.31i)T \)
good3 \( 1 - 1.34iT - 3T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 - 1.57T + 13T^{2} \)
17 \( 1 + 1.11iT - 17T^{2} \)
19 \( 1 - 8.44iT - 19T^{2} \)
23 \( 1 - 2.62iT - 23T^{2} \)
29 \( 1 + 3.43iT - 29T^{2} \)
31 \( 1 - 9.70T + 31T^{2} \)
37 \( 1 + 6.22iT - 37T^{2} \)
41 \( 1 + 3.13iT - 41T^{2} \)
43 \( 1 - 7.45T + 43T^{2} \)
47 \( 1 + 9.40T + 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + 6.16iT - 59T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 + 2.15T + 67T^{2} \)
71 \( 1 - 7.87iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + 7.70iT - 79T^{2} \)
83 \( 1 + 0.813iT - 83T^{2} \)
89 \( 1 + 5.12iT - 89T^{2} \)
97 \( 1 - 0.833iT - 97T^{2} \)
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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

−11.63327883455462513850144653665, −10.73546962654083636866204944370, −10.09238412461873253250083240566, −9.250007319442169873053097660003, −8.091096636445103749918274517255, −7.39115742185412529694097861406, −6.14579865738885074173946261328, −4.33193389193378146768906909001, −3.55290489740272365834407716371, −1.38286089536520033667300948262, 1.22581732593017370494920754283, 2.67492569259572910501900793121, 4.74978445691751095481123209056, 6.32522617041897792437091259918, 6.91955241898254164254939606261, 8.088158857321060821768926759917, 8.704463467718625739645057139267, 9.699332369162429696400091232549, 10.99170195832984337975588413273, 11.68798037332585172915082383333

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