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Properties

Label 2-280-56.27-c1-0-16
Degree $2$
Conductor $280$
Sign $0.585 - 0.810i$
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  + (1.14 + 0.833i)2-s − 0.586i·3-s + (0.609 + 1.90i)4-s − 5-s + (0.489 − 0.670i)6-s + (2.52 + 0.792i)7-s + (−0.892 + 2.68i)8-s + 2.65·9-s + (−1.14 − 0.833i)10-s − 0.580·11-s + (1.11 − 0.357i)12-s − 1.14·13-s + (2.22 + 3.00i)14-s + 0.586i·15-s + (−3.25 + 2.32i)16-s − 1.82i·17-s + ⋯
L(s)  = 1  + (0.807 + 0.589i)2-s − 0.338i·3-s + (0.304 + 0.952i)4-s − 0.447·5-s + (0.199 − 0.273i)6-s + (0.954 + 0.299i)7-s + (−0.315 + 0.948i)8-s + 0.885·9-s + (−0.361 − 0.263i)10-s − 0.174·11-s + (0.322 − 0.103i)12-s − 0.318·13-s + (0.594 + 0.804i)14-s + 0.151i·15-s + (−0.814 + 0.580i)16-s − 0.443i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.585 - 0.810i$
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: \((2,\ 280,\ (\ :1/2),\ 0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78350 + 0.912264i\)
\(L(\frac12)\) \(\approx\) \(1.78350 + 0.912264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 0.833i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.52 - 0.792i)T \)
good3 \( 1 + 0.586iT - 3T^{2} \)
11 \( 1 + 0.580T + 11T^{2} \)
13 \( 1 + 1.14T + 13T^{2} \)
17 \( 1 + 1.82iT - 17T^{2} \)
19 \( 1 - 4.72iT - 19T^{2} \)
23 \( 1 + 2.79iT - 23T^{2} \)
29 \( 1 + 7.40iT - 29T^{2} \)
31 \( 1 + 4.73T + 31T^{2} \)
37 \( 1 + 4.35iT - 37T^{2} \)
41 \( 1 + 8.46iT - 41T^{2} \)
43 \( 1 + 4.32T + 43T^{2} \)
47 \( 1 + 3.56T + 47T^{2} \)
53 \( 1 + 5.48iT - 53T^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 + 0.275T + 61T^{2} \)
67 \( 1 + 7.71T + 67T^{2} \)
71 \( 1 - 1.13iT - 71T^{2} \)
73 \( 1 - 1.78iT - 73T^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 - 15.3iT - 89T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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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

−12.14855411570206003939181528647, −11.46277871427293060529698139437, −10.26620486720946288252089502857, −8.763413294978573400180120809876, −7.78405663412280673403323951887, −7.23953518227668588663363638777, −5.94291650872449340757348947000, −4.83708453933848015950593799346, −3.88639840565343733759056934538, −2.15311390228788829771341416382, 1.58020663266535232613655993611, 3.33663684183036563588119837099, 4.51940868679729174658461000194, 5.11984731981318697711666032386, 6.73259230249439209566442200396, 7.67853471912939604109680534867, 9.082652023619267858768508690681, 10.16319581105440182329325751569, 10.94959028883670165845410466789, 11.61823163319546702166834481133

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