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Properties

Label 2-280-56.27-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.378 - 0.925i$
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  + (−0.275 − 1.38i)2-s + 3.19i·3-s + (−1.84 + 0.764i)4-s − 5-s + (4.43 − 0.881i)6-s + (2.59 + 0.525i)7-s + (1.56 + 2.35i)8-s − 7.23·9-s + (0.275 + 1.38i)10-s − 3.34·11-s + (−2.44 − 5.91i)12-s − 3.90·13-s + (0.0143 − 3.74i)14-s − 3.19i·15-s + (2.83 − 2.82i)16-s + 2.92i·17-s + ⋯
L(s)  = 1  + (−0.194 − 0.980i)2-s + 1.84i·3-s + (−0.924 + 0.382i)4-s − 0.447·5-s + (1.81 − 0.359i)6-s + (0.980 + 0.198i)7-s + (0.555 + 0.831i)8-s − 2.41·9-s + (0.0871 + 0.438i)10-s − 1.00·11-s + (−0.706 − 1.70i)12-s − 1.08·13-s + (0.00384 − 0.999i)14-s − 0.826i·15-s + (0.707 − 0.706i)16-s + 0.710i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.378 - 0.925i$
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: \((2,\ 280,\ (\ :1/2),\ -0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409852 + 0.610542i\)
\(L(\frac12)\) \(\approx\) \(0.409852 + 0.610542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.275 + 1.38i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.59 - 0.525i)T \)
good3 \( 1 - 3.19iT - 3T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
13 \( 1 + 3.90T + 13T^{2} \)
17 \( 1 - 2.92iT - 17T^{2} \)
19 \( 1 - 6.33iT - 19T^{2} \)
23 \( 1 + 3.44iT - 23T^{2} \)
29 \( 1 - 2.68iT - 29T^{2} \)
31 \( 1 - 2.52T + 31T^{2} \)
37 \( 1 - 4.70iT - 37T^{2} \)
41 \( 1 - 5.59iT - 41T^{2} \)
43 \( 1 - 8.62T + 43T^{2} \)
47 \( 1 - 0.506T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + 0.802iT - 59T^{2} \)
61 \( 1 - 7.97T + 61T^{2} \)
67 \( 1 - 6.37T + 67T^{2} \)
71 \( 1 + 1.11iT - 71T^{2} \)
73 \( 1 - 5.91iT - 73T^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 - 4.29iT - 83T^{2} \)
89 \( 1 - 2.00iT - 89T^{2} \)
97 \( 1 + 6.06iT - 97T^{2} \)
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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

−11.80851669612403467958846923002, −10.98221277780955824900474478256, −10.31534730317571453887044440209, −9.716985055623094869182762335622, −8.458460691003621452756958371505, −8.013660901274664696578939568573, −5.42795107656440439047853858857, −4.72613994954225622835504437763, −3.83682759349115567527122770241, −2.56110443238250942690002611830, 0.59145645816036625309977948341, 2.44389462598731138029127812881, 4.78794207252563491806499694911, 5.71085727118519963956387388942, 7.20686325611393577248582832359, 7.37393958236631042697881211693, 8.180911400120443746527884800327, 9.185732337679137369994027974917, 10.80694563121406827317220095329, 11.78400845050905907268729568847

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