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Properties

Label 2-280-280.13-c1-0-43
Degree $2$
Conductor $280$
Sign $-0.904 - 0.425i$
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 − i)2-s + (−1.45 − 1.45i)3-s − 2i·4-s + (−2.22 + 0.254i)5-s − 2.91·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 1.25i·9-s + (−1.96 + 2.47i)10-s + (−2.91 + 2.91i)12-s + (1.45 + 1.45i)13-s + 3.74i·14-s + (3.61 + 2.87i)15-s − 4·16-s + (1.25 + 1.25i)18-s − 8.37i·19-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.842 − 0.842i)3-s − i·4-s + (−0.993 + 0.113i)5-s − 1.19·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 0.419i·9-s + (−0.622 + 0.782i)10-s + (−0.842 + 0.842i)12-s + (0.404 + 0.404i)13-s + 0.999i·14-s + (0.932 + 0.741i)15-s − 16-s + (0.296 + 0.296i)18-s − 1.92i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.904 - 0.425i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.904 - 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144014 + 0.644506i\)
\(L(\frac12)\) \(\approx\) \(0.144014 + 0.644506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (2.22 - 0.254i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (1.45 + 1.45i)T + 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1.45 - 1.45i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 8.37iT - 19T^{2} \)
23 \( 1 + (6.74 + 6.74i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 11.2iT - 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 8.25iT - 79T^{2} \)
83 \( 1 + (-4.00 - 4.00i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−11.49737859298670229560380592404, −10.95873127328900183436553135950, −9.607174097427228908276786286691, −8.485353350839848772712164581294, −6.83962344429728884361163193312, −6.41089728611544978558545274374, −5.15208992771478852450653058733, −3.88464178802864069036729086043, −2.48052472986942255065423867360, −0.43010155017251591624800212367, 3.64150647869528983291310787503, 4.04075786325345303723552738706, 5.37307288625315374445527250212, 6.19928376883660876555274863135, 7.49413322136837161238660047283, 8.200320344005816268690288294916, 9.735207507991461113053090730009, 10.63358459846950997989642221122, 11.63472959310115890823884884823, 12.28553919174969553778006528042

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