Properties

Label 2-280-56.19-c1-0-28
Degree $2$
Conductor $280$
Sign $-0.951 + 0.307i$
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  + (1.03 − 0.961i)2-s + (−2.66 + 1.54i)3-s + (0.149 − 1.99i)4-s + (0.5 − 0.866i)5-s + (−1.28 + 4.16i)6-s + (−2.53 + 0.754i)7-s + (−1.76 − 2.21i)8-s + (3.24 − 5.61i)9-s + (−0.314 − 1.37i)10-s + (−1.64 − 2.84i)11-s + (2.67 + 5.55i)12-s − 6.72·13-s + (−1.90 + 3.22i)14-s + 3.08i·15-s + (−3.95 − 0.595i)16-s + (3.32 − 1.91i)17-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (−1.54 + 0.889i)3-s + (0.0746 − 0.997i)4-s + (0.223 − 0.387i)5-s + (−0.524 + 1.69i)6-s + (−0.958 + 0.285i)7-s + (−0.623 − 0.781i)8-s + (1.08 − 1.87i)9-s + (−0.0995 − 0.435i)10-s + (−0.495 − 0.857i)11-s + (0.771 + 1.60i)12-s − 1.86·13-s + (−0.508 + 0.860i)14-s + 0.795i·15-s + (−0.988 − 0.148i)16-s + (0.806 − 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.951 + 0.307i$
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: \((2,\ 280,\ (\ :1/2),\ -0.951 + 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0768393 - 0.488445i\)
\(L(\frac12)\) \(\approx\) \(0.0768393 - 0.488445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 + 0.961i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.53 - 0.754i)T \)
good3 \( 1 + (2.66 - 1.54i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.64 + 2.84i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.72T + 13T^{2} \)
17 \( 1 + (-3.32 + 1.91i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.618 + 0.357i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.09 + 0.631i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.04iT - 29T^{2} \)
31 \( 1 + (0.0335 + 0.0581i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.498 - 0.287i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.230iT - 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (-4.23 + 7.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.16 - 1.25i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.986 + 0.569i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0888 + 0.153i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 + 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (3.89 - 2.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.66 + 5.58i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.24iT - 83T^{2} \)
89 \( 1 + (13.3 + 7.68i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.76iT - 97T^{2} \)
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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

−11.56782745766841174698174990245, −10.51411544219968522713549037518, −9.941873852643475851486262623197, −9.229654368625420761602939971055, −7.00861732677098126294384089669, −5.80090936834385774460564133334, −5.35649282153875178606630466951, −4.33876202348124406561286737008, −2.92659300722153648664985228775, −0.33227754752332076960505813890, 2.50571879838100401361672478933, 4.44778215589710446196722743885, 5.52714262470188427465391413521, 6.25764777793592863145904053071, 7.29725178626296755026189283286, 7.54536130639853060933186020952, 9.713242606040028503125314357374, 10.54461197477493474691039236575, 11.78749295911419829221036832817, 12.54951945387528753513863410747

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