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Properties

Label 2-280-56.27-c1-0-8
Degree $2$
Conductor $280$
Sign $-0.879 - 0.476i$
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  + (1.07 + 0.924i)2-s + 2.99i·3-s + (0.291 + 1.97i)4-s − 5-s + (−2.77 + 3.20i)6-s + (−0.183 − 2.63i)7-s + (−1.51 + 2.38i)8-s − 5.98·9-s + (−1.07 − 0.924i)10-s + 4.87·11-s + (−5.93 + 0.874i)12-s + 2.42·13-s + (2.24 − 2.99i)14-s − 2.99i·15-s + (−3.82 + 1.15i)16-s + 3.92i·17-s + ⋯
L(s)  = 1  + (0.756 + 0.653i)2-s + 1.73i·3-s + (0.145 + 0.989i)4-s − 0.447·5-s + (−1.13 + 1.30i)6-s + (−0.0693 − 0.997i)7-s + (−0.536 + 0.844i)8-s − 1.99·9-s + (−0.338 − 0.292i)10-s + 1.47·11-s + (−1.71 + 0.252i)12-s + 0.673·13-s + (0.599 − 0.800i)14-s − 0.773i·15-s + (−0.957 + 0.288i)16-s + 0.953i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.879 - 0.476i$
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: \((2,\ 280,\ (\ :1/2),\ -0.879 - 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431702 + 1.70333i\)
\(L(\frac12)\) \(\approx\) \(0.431702 + 1.70333i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 - 0.924i)T \)
5 \( 1 + T \)
7 \( 1 + (0.183 + 2.63i)T \)
good3 \( 1 - 2.99iT - 3T^{2} \)
11 \( 1 - 4.87T + 11T^{2} \)
13 \( 1 - 2.42T + 13T^{2} \)
17 \( 1 - 3.92iT - 17T^{2} \)
19 \( 1 + 5.24iT - 19T^{2} \)
23 \( 1 - 0.114iT - 23T^{2} \)
29 \( 1 - 3.60iT - 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 7.83iT - 37T^{2} \)
41 \( 1 + 10.4iT - 41T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 0.668iT - 53T^{2} \)
59 \( 1 + 1.37iT - 59T^{2} \)
61 \( 1 - 1.17T + 61T^{2} \)
67 \( 1 - 2.57T + 67T^{2} \)
71 \( 1 + 9.43iT - 71T^{2} \)
73 \( 1 + 5.62iT - 73T^{2} \)
79 \( 1 + 11.8iT - 79T^{2} \)
83 \( 1 + 5.52iT - 83T^{2} \)
89 \( 1 - 6.21iT - 89T^{2} \)
97 \( 1 - 7.85iT - 97T^{2} \)
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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

−12.16467330674288662999024731213, −11.15423495581493117574104257931, −10.61682866261414320629834995183, −9.216004495815148609062599518732, −8.619235006754665766486591016555, −7.20108506426679777887002539812, −6.14052322128408034061577253161, −4.82843089137864225572452545151, −3.96735406976190994795067358711, −3.49990867872448643341828492070, 1.23339094645801521401980843045, 2.47067186347300516991029382990, 3.84924903189495494815463833083, 5.65733801016444741312277771787, 6.32220542532044251052773798506, 7.30952864155726517906487711989, 8.563740147975275942246861803331, 9.470331939769382315651753736207, 11.21497715723167280827305655050, 11.76067632955982790271423736372

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