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Properties

Label 2-280-56.3-c1-0-20
Degree $2$
Conductor $280$
Sign $0.657 + 0.753i$
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  + (1.39 + 0.252i)2-s + (−1.84 − 1.06i)3-s + (1.87 + 0.702i)4-s + (−0.5 − 0.866i)5-s + (−2.29 − 1.94i)6-s + (2.17 − 1.50i)7-s + (2.42 + 1.45i)8-s + (0.759 + 1.31i)9-s + (−0.477 − 1.33i)10-s + (2.04 − 3.54i)11-s + (−2.70 − 3.28i)12-s − 4.95·13-s + (3.40 − 1.54i)14-s + 2.12i·15-s + (3.01 + 2.63i)16-s + (2.09 + 1.20i)17-s + ⋯
L(s)  = 1  + (0.983 + 0.178i)2-s + (−1.06 − 0.613i)3-s + (0.936 + 0.351i)4-s + (−0.223 − 0.387i)5-s + (−0.936 − 0.793i)6-s + (0.822 − 0.569i)7-s + (0.858 + 0.512i)8-s + (0.253 + 0.438i)9-s + (−0.150 − 0.420i)10-s + (0.616 − 1.06i)11-s + (−0.779 − 0.948i)12-s − 1.37·13-s + (0.910 − 0.413i)14-s + 0.548i·15-s + (0.753 + 0.657i)16-s + (0.507 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56558 - 0.711924i\)
\(L(\frac12)\) \(\approx\) \(1.56558 - 0.711924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.252i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.17 + 1.50i)T \)
good3 \( 1 + (1.84 + 1.06i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.04 + 3.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.95T + 13T^{2} \)
17 \( 1 + (-2.09 - 1.20i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.22 + 3.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.443 + 0.255i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.08iT - 29T^{2} \)
31 \( 1 + (2.30 - 3.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.64 - 4.99i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.81iT - 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (0.698 + 1.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.79 - 3.92i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.82 - 5.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.33 - 2.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 6.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (-6.99 - 4.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.21 - 1.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.94iT - 83T^{2} \)
89 \( 1 + (3.81 - 2.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.67iT - 97T^{2} \)
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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

−11.73059225097184097020594623550, −11.40212375697537573625162691268, −10.26174081972508515286191413911, −8.548975176007914099866826829400, −7.37962429161790135890361219951, −6.75432997932303640352798716757, −5.43109220516712219832262723106, −4.91414333405468079253097293667, −3.39161122396291120893149182654, −1.27326040874618914099030431643, 2.15099859847186321655124607172, 3.88103931489052048125966912542, 5.05700069588666873760522899379, 5.44464146140767729598993255109, 6.86569692593148428722916236153, 7.73161090675072995508085979702, 9.678033449739943709553389145241, 10.28869708859096920700193834213, 11.45730421238423390133790654515, 11.87601129187810414391267619738

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