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Properties

Label 2-280-56.19-c1-0-31
Degree $2$
Conductor $280$
Sign $-0.366 + 0.930i$
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  + (0.743 − 1.20i)2-s + (1.94 − 1.12i)3-s + (−0.893 − 1.78i)4-s + (0.5 − 0.866i)5-s + (0.0964 − 3.17i)6-s + (−2.47 + 0.947i)7-s + (−2.81 − 0.257i)8-s + (1.02 − 1.78i)9-s + (−0.669 − 1.24i)10-s + (0.656 + 1.13i)11-s + (−3.75 − 2.48i)12-s + 3.02·13-s + (−0.697 + 3.67i)14-s − 2.24i·15-s + (−2.40 + 3.19i)16-s + (−0.313 + 0.181i)17-s + ⋯
L(s)  = 1  + (0.526 − 0.850i)2-s + (1.12 − 0.649i)3-s + (−0.446 − 0.894i)4-s + (0.223 − 0.387i)5-s + (0.0393 − 1.29i)6-s + (−0.933 + 0.358i)7-s + (−0.995 − 0.0908i)8-s + (0.342 − 0.593i)9-s + (−0.211 − 0.393i)10-s + (0.198 + 0.342i)11-s + (−1.08 − 0.716i)12-s + 0.838·13-s + (−0.186 + 0.982i)14-s − 0.580i·15-s + (−0.601 + 0.799i)16-s + (−0.0760 + 0.0439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.366 + 0.930i$
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: \((2,\ 280,\ (\ :1/2),\ -0.366 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17959 - 1.73184i\)
\(L(\frac12)\) \(\approx\) \(1.17959 - 1.73184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.743 + 1.20i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.47 - 0.947i)T \)
good3 \( 1 + (-1.94 + 1.12i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.656 - 1.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
17 \( 1 + (0.313 - 0.181i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.16 - 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.74 - 3.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.35iT - 29T^{2} \)
31 \( 1 + (3.37 + 5.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.89 + 2.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.07iT - 41T^{2} \)
43 \( 1 + 5.01T + 43T^{2} \)
47 \( 1 + (6.23 - 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.39 + 1.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.5 - 6.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.04 - 8.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.897 + 1.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (-7.83 + 4.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.89 - 4.55i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 + (1.99 + 1.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.74iT - 97T^{2} \)
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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

−11.85449662086555875884983869127, −10.67520019299447707359856686606, −9.320562273809260692231639518813, −9.177342401603390082236761682566, −7.82623099935774139852840811531, −6.49504173461819973524751942282, −5.39772992516266861699225352342, −3.77909542295366769931951236012, −2.84459284814505518381570881332, −1.54980266288710713676115878910, 3.12212137160322493311049514054, 3.51493102527346806238505751809, 4.95358159147546436324237585366, 6.36770159347953575861486331125, 7.11218574138667195347192055510, 8.458873244397621808122462911079, 9.054843475826737368069887420446, 9.960268576813237606806953244135, 11.13929612798297253400246540575, 12.54350494474257700016605334006

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