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Properties

Label 2-280-280.13-c1-0-22
Degree $2$
Conductor $280$
Sign $-0.425 + 0.904i$
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 + (−1.96 − 1.96i)3-s − 2i·4-s + (0.254 + 2.22i)5-s + 3.93·6-s + (1.87 − 1.87i)7-s + (2 + 2i)8-s + 4.74i·9-s + (−2.47 − 1.96i)10-s + (−3.93 + 3.93i)12-s + (−4.88 − 4.88i)13-s + 3.74i·14-s + (3.87 − 4.87i)15-s − 4·16-s + (−4.74 − 4.74i)18-s − 2.41i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.13 − 1.13i)3-s − i·4-s + (0.113 + 0.993i)5-s + 1.60·6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + 1.58i·9-s + (−0.782 − 0.622i)10-s + (−1.13 + 1.13i)12-s + (−1.35 − 1.35i)13-s + 0.999i·14-s + (0.999 − 1.25i)15-s − 16-s + (−1.11 − 1.11i)18-s − 0.552i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.425 + 0.904i$
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: \((2,\ 280,\ (\ :1/2),\ -0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203948 - 0.321318i\)
\(L(\frac12)\) \(\approx\) \(0.203948 - 0.321318i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.254 - 2.22i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good3 \( 1 + (1.96 + 1.96i)T + 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (4.88 + 4.88i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
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 \( 1 + 10.4iT - 59T^{2} \)
61 \( 1 - 6.47T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 8.25iT - 79T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \)
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

−11.35739815101660782289373823287, −10.57236959295210381426847516476, −10.00857269745167477748475604830, −8.114249422796344418749016276499, −7.48721772365423079037987633257, −6.78873115816975860768353945666, −5.89015535993968206859346902592, −4.85823112195092454801929660205, −2.19344225615377220546993600807, −0.40447727518961596242918048311, 1.87473191363303879610551512204, 4.08117541941384295050310182594, 4.82585642517749726050328896500, 5.84998599718303763183048229582, 7.56365737628655576133914653984, 8.774553588622529115870742311648, 9.573532137981231989449297832325, 10.10648886224627930277608064344, 11.41146180890839939676215005553, 11.85752877594318138977411536887

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