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Properties

Label 2-280-280.139-c1-0-40
Degree $2$
Conductor $280$
Sign $-0.170 + 0.985i$
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  + (0.244 − 1.39i)2-s + 2.91·3-s + (−1.88 − 0.682i)4-s + (−1.83 − 1.27i)5-s + (0.712 − 4.05i)6-s + (1.52 − 2.16i)7-s + (−1.41 + 2.45i)8-s + 5.47·9-s + (−2.22 + 2.24i)10-s − 0.0929·11-s + (−5.47 − 1.98i)12-s + 4.08i·13-s + (−2.64 − 2.65i)14-s + (−5.34 − 3.71i)15-s + (3.06 + 2.56i)16-s − 4.24·17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + 1.68·3-s + (−0.940 − 0.341i)4-s + (−0.821 − 0.570i)5-s + (0.290 − 1.65i)6-s + (0.575 − 0.817i)7-s + (−0.498 + 0.866i)8-s + 1.82·9-s + (−0.704 + 0.710i)10-s − 0.0280·11-s + (−1.57 − 0.573i)12-s + 1.13i·13-s + (−0.705 − 0.708i)14-s + (−1.38 − 0.958i)15-s + (0.767 + 0.641i)16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.170 + 0.985i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.170 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24100 - 1.47454i\)
\(L(\frac12)\) \(\approx\) \(1.24100 - 1.47454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.244 + 1.39i)T \)
5 \( 1 + (1.83 + 1.27i)T \)
7 \( 1 + (-1.52 + 2.16i)T \)
good3 \( 1 - 2.91T + 3T^{2} \)
11 \( 1 + 0.0929T + 11T^{2} \)
13 \( 1 - 4.08iT - 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + 2.39iT - 19T^{2} \)
23 \( 1 - 4.52T + 23T^{2} \)
29 \( 1 - 4.35iT - 29T^{2} \)
31 \( 1 + 1.10T + 31T^{2} \)
37 \( 1 - 8.54T + 37T^{2} \)
41 \( 1 + 6.10iT - 41T^{2} \)
43 \( 1 - 4.60iT - 43T^{2} \)
47 \( 1 - 7.93iT - 47T^{2} \)
53 \( 1 + 14.3T + 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 + 1.92T + 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + 9.75iT - 71T^{2} \)
73 \( 1 - 6.19T + 73T^{2} \)
79 \( 1 + 3.42iT - 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 4.46iT - 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−11.48364950676100518723470887514, −10.82403167142048335885326383724, −9.382949880400120134613132418798, −8.943778688187578796228194410466, −8.061043477778999964531301912688, −7.10174535505068992129311305586, −4.61782868500609991583021869136, −4.16776428349304374592746600238, −2.94301989653195320117947202844, −1.50997330028077935368188133660, 2.65495786270740955259410004721, 3.67478717093312678683585569388, 4.85896504413441160440903220453, 6.42163121315409963759064808407, 7.67930304149879988339504542831, 8.079187333850646927424222103877, 8.840673131867325391647175436658, 9.799274088872167000884294772600, 11.18337298319427330011304777671, 12.53648283587783789578193562088

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