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Properties

Label 2-280-280.107-c1-0-5
Degree $2$
Conductor $280$
Sign $0.910 + 0.412i$
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  + (−0.786 − 1.17i)2-s + (−2.51 − 0.673i)3-s + (−0.762 + 1.84i)4-s + (−2.09 − 0.774i)5-s + (1.18 + 3.48i)6-s + (0.419 + 2.61i)7-s + (2.77 − 0.559i)8-s + (3.26 + 1.88i)9-s + (0.740 + 3.07i)10-s + (0.673 + 1.16i)11-s + (3.16 − 4.13i)12-s + (−3.02 − 3.02i)13-s + (2.73 − 2.54i)14-s + (4.75 + 3.35i)15-s + (−2.83 − 2.81i)16-s + (1.09 − 4.08i)17-s + ⋯
L(s)  = 1  + (−0.556 − 0.830i)2-s + (−1.45 − 0.388i)3-s + (−0.381 + 0.924i)4-s + (−0.938 − 0.346i)5-s + (0.484 + 1.42i)6-s + (0.158 + 0.987i)7-s + (0.980 − 0.197i)8-s + (1.08 + 0.628i)9-s + (0.234 + 0.972i)10-s + (0.202 + 0.351i)11-s + (0.912 − 1.19i)12-s + (−0.840 − 0.840i)13-s + (0.732 − 0.681i)14-s + (1.22 + 0.867i)15-s + (−0.709 − 0.704i)16-s + (0.265 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424410 - 0.0916564i\)
\(L(\frac12)\) \(\approx\) \(0.424410 - 0.0916564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.786 + 1.17i)T \)
5 \( 1 + (2.09 + 0.774i)T \)
7 \( 1 + (-0.419 - 2.61i)T \)
good3 \( 1 + (2.51 + 0.673i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.673 - 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.02 + 3.02i)T + 13iT^{2} \)
17 \( 1 + (-1.09 + 4.08i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.52 - 3.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.25 + 0.605i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.41T + 29T^{2} \)
31 \( 1 + (5.73 - 3.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.39 - 8.95i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.58T + 41T^{2} \)
43 \( 1 + (-3.73 + 3.73i)T - 43iT^{2} \)
47 \( 1 + (-0.998 - 3.72i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.0240 + 0.0898i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.663 + 0.383i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.18 - 0.685i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.24 - 8.38i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.47iT - 71T^{2} \)
73 \( 1 + (-0.914 - 0.245i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.50 + 2.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.12 - 7.12i)T - 83iT^{2} \)
89 \( 1 + (3.89 + 2.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 10.9i)T + 97iT^{2} \)
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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

−11.83096090627595978512522848049, −11.20247443422602071506901952408, −10.08987074301952835989964029600, −9.108293457603331046446854665545, −7.85007234538653968506081502634, −7.16479881876769410417692927315, −5.49133366485048371420662376470, −4.74735523569959197101229157756, −3.00267580544787415027431875068, −0.985716996776189009481858385463, 0.67141377096373478318676541653, 4.04684837287439243710008825915, 4.90766156538241224218215435513, 6.05851274196807212781049472678, 7.11832365076719984035039415258, 7.62363835868531137790123025220, 9.155670577459638278057565533677, 10.16052122248116894019433849119, 11.07288262638735874860258963918, 11.40603345575499933384510190638

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