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Properties

Label 2-280-56.27-c1-0-4
Degree $2$
Conductor $280$
Sign $-0.142 - 0.989i$
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  + (−1.38 + 0.303i)2-s + 1.34i·3-s + (1.81 − 0.837i)4-s + 5-s + (−0.406 − 1.85i)6-s + (−1.28 + 2.31i)7-s + (−2.25 + 1.70i)8-s + 1.20·9-s + (−1.38 + 0.303i)10-s + 2.44·11-s + (1.12 + 2.43i)12-s − 1.57·13-s + (1.06 − 3.58i)14-s + 1.34i·15-s + (2.59 − 3.04i)16-s − 1.11i·17-s + ⋯
L(s)  = 1  + (−0.976 + 0.214i)2-s + 0.774i·3-s + (0.908 − 0.418i)4-s + 0.447·5-s + (−0.166 − 0.756i)6-s + (−0.483 + 0.875i)7-s + (−0.797 + 0.603i)8-s + 0.400·9-s + (−0.436 + 0.0958i)10-s + 0.738·11-s + (0.324 + 0.703i)12-s − 0.435·13-s + (0.284 − 0.958i)14-s + 0.346i·15-s + (0.649 − 0.760i)16-s − 0.271i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.142 - 0.989i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569583 + 0.657603i\)
\(L(\frac12)\) \(\approx\) \(0.569583 + 0.657603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.303i)T \)
5 \( 1 - T \)
7 \( 1 + (1.28 - 2.31i)T \)
good3 \( 1 - 1.34iT - 3T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 1.57T + 13T^{2} \)
17 \( 1 + 1.11iT - 17T^{2} \)
19 \( 1 - 8.44iT - 19T^{2} \)
23 \( 1 + 2.62iT - 23T^{2} \)
29 \( 1 - 3.43iT - 29T^{2} \)
31 \( 1 + 9.70T + 31T^{2} \)
37 \( 1 - 6.22iT - 37T^{2} \)
41 \( 1 + 3.13iT - 41T^{2} \)
43 \( 1 - 7.45T + 43T^{2} \)
47 \( 1 - 9.40T + 47T^{2} \)
53 \( 1 + 11.6iT - 53T^{2} \)
59 \( 1 + 6.16iT - 59T^{2} \)
61 \( 1 - 9.44T + 61T^{2} \)
67 \( 1 + 2.15T + 67T^{2} \)
71 \( 1 + 7.87iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 - 7.70iT - 79T^{2} \)
83 \( 1 + 0.813iT - 83T^{2} \)
89 \( 1 + 5.12iT - 89T^{2} \)
97 \( 1 - 0.833iT - 97T^{2} \)
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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

−12.01899212259584449391584649810, −10.79970708500516692860007082435, −9.970736857857461941879881645688, −9.395753246441666162513063117294, −8.618675618969095812454077153849, −7.29519388739031493769466712276, −6.21786107808791704218221489415, −5.27716517643229262787136656236, −3.55372481424725496181667774466, −1.91233050535756757058563382852, 0.954027433515987197822461561712, 2.40234878092203970455099851567, 4.03956189559516934302546570895, 6.03386043667015101428790968242, 7.18345737837795445918899795206, 7.33003990833043051680772368879, 8.958537305499851448459886028737, 9.591073892701813253505903048553, 10.59856449152034546376727004172, 11.43858983567817227103145982454

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