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Properties

Label 2-280-56.19-c1-0-3
Degree $2$
Conductor $280$
Sign $-0.995 - 0.0931i$
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  + (0.542 + 1.30i)2-s + (−0.908 + 0.524i)3-s + (−1.41 + 1.41i)4-s + (−0.5 + 0.866i)5-s + (−1.17 − 0.901i)6-s + (2.14 + 1.54i)7-s + (−2.61 − 1.07i)8-s + (−0.949 + 1.64i)9-s + (−1.40 − 0.183i)10-s + (−1.17 − 2.03i)11-s + (0.539 − 2.02i)12-s − 1.21·13-s + (−0.852 + 3.64i)14-s − 1.04i·15-s + (−0.0131 − 3.99i)16-s + (−4.23 + 2.44i)17-s + ⋯
L(s)  = 1  + (0.383 + 0.923i)2-s + (−0.524 + 0.302i)3-s + (−0.705 + 0.708i)4-s + (−0.223 + 0.387i)5-s + (−0.480 − 0.368i)6-s + (0.811 + 0.583i)7-s + (−0.924 − 0.380i)8-s + (−0.316 + 0.548i)9-s + (−0.443 − 0.0580i)10-s + (−0.354 − 0.614i)11-s + (0.155 − 0.585i)12-s − 0.336·13-s + (−0.227 + 0.973i)14-s − 0.270i·15-s + (−0.00329 − 0.999i)16-s + (−1.02 + 0.593i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.995 - 0.0931i$
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: \((2,\ 280,\ (\ :1/2),\ -0.995 - 0.0931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0459132 + 0.984125i\)
\(L(\frac12)\) \(\approx\) \(0.0459132 + 0.984125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.542 - 1.30i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.14 - 1.54i)T \)
good3 \( 1 + (0.908 - 0.524i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.17 + 2.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.21T + 13T^{2} \)
17 \( 1 + (4.23 - 2.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.21 + 1.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.59 - 4.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.21iT - 29T^{2} \)
31 \( 1 + (-1.68 - 2.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.16 - 3.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.18iT - 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + (-5.01 + 8.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.03 - 4.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.90 - 5.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.560 + 0.971i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.386 - 0.670i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (-11.1 + 6.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.34 + 0.776i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.47iT - 83T^{2} \)
89 \( 1 + (9.58 + 5.53i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−12.37212301236539998039301191075, −11.24242231415885586809617285095, −10.79259755580235978566242517612, −9.132416682392407895892897167880, −8.354176840141006994002062292746, −7.40604938217344765151824111647, −6.21663334516424337959374205239, −5.26971871074079520110135703074, −4.49301540546185018711700289457, −2.82325402644859635706226202630, 0.71694746406897087027042682139, 2.44667610538007382741272956304, 4.23472212867131225621117323336, 4.91747738516859378159747472403, 6.20299262824762510102788269002, 7.47155924761536272534037823050, 8.751716230389489296534780640056, 9.642686031110321455220575605157, 10.99783771254326121306837635966, 11.22537849323148905449361429217

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