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Properties

Label 2-280-35.4-c1-0-8
Degree $2$
Conductor $280$
Sign $0.823 + 0.566i$
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  + (0.277 + 0.159i)3-s + (1.85 − 1.24i)5-s + (1.50 − 2.17i)7-s + (−1.44 − 2.50i)9-s + (−2.08 + 3.60i)11-s − 2.89i·13-s + (0.713 − 0.0477i)15-s + (3.72 + 2.15i)17-s + (0.979 + 1.69i)19-s + (0.765 − 0.361i)21-s + (2.23 − 1.28i)23-s + (1.90 − 4.62i)25-s − 1.88i·27-s + 5.96·29-s + (−4.71 + 8.16i)31-s + ⋯
L(s)  = 1  + (0.159 + 0.0923i)3-s + (0.830 − 0.556i)5-s + (0.569 − 0.821i)7-s + (−0.482 − 0.836i)9-s + (−0.628 + 1.08i)11-s − 0.803i·13-s + (0.184 − 0.0123i)15-s + (0.903 + 0.521i)17-s + (0.224 + 0.389i)19-s + (0.167 − 0.0788i)21-s + (0.465 − 0.268i)23-s + (0.380 − 0.924i)25-s − 0.363i·27-s + 1.10·29-s + (−0.846 + 1.46i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45284 - 0.451417i\)
\(L(\frac12)\) \(\approx\) \(1.45284 - 0.451417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.85 + 1.24i)T \)
7 \( 1 + (-1.50 + 2.17i)T \)
good3 \( 1 + (-0.277 - 0.159i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.08 - 3.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.89iT - 13T^{2} \)
17 \( 1 + (-3.72 - 2.15i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.979 - 1.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.23 + 1.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
31 \( 1 + (4.71 - 8.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.48 - 3.16i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.19T + 41T^{2} \)
43 \( 1 - 4.73iT - 43T^{2} \)
47 \( 1 + (3.84 - 2.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.37 - 3.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.61 - 2.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.94 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.58T + 71T^{2} \)
73 \( 1 + (8.57 + 4.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.75iT - 83T^{2} \)
89 \( 1 + (1.50 + 2.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.414iT - 97T^{2} \)
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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

−12.00024448075500368783658162970, −10.37791612997851892391506521139, −10.16695232751822128344943202266, −8.882858866742404449784507630582, −8.009819123709668671303420939066, −6.86985597816139704715375528970, −5.57683980590968067285054746728, −4.68587538613849806717047064087, −3.16450317288185279406618341869, −1.37782129296419317576502478756, 2.07226669144535136298005278263, 3.09943241864392631063698567372, 5.17389060144370071435785867397, 5.71696237935709603853143803350, 7.06818067294858736699232432238, 8.213475257404739491217169381315, 9.008672964573302033258969790207, 10.10477934058339758334309066228, 11.12835666169957958499481113007, 11.67716332750540796205154125976

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