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Properties

Label 2-280-280.19-c1-0-9
Degree $2$
Conductor $280$
Sign $-0.436 - 0.899i$
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  + (−1.02 + 0.971i)2-s + (1.39 + 2.42i)3-s + (0.113 − 1.99i)4-s + (2.07 − 0.823i)5-s + (−3.78 − 1.13i)6-s + (−2.32 − 1.26i)7-s + (1.82 + 2.16i)8-s + (−2.41 + 4.17i)9-s + (−1.33 + 2.86i)10-s + (2.81 + 4.87i)11-s + (4.99 − 2.51i)12-s + 3.09i·13-s + (3.61 − 0.955i)14-s + (4.90 + 3.88i)15-s + (−3.97 − 0.454i)16-s + (−1.34 − 2.32i)17-s + ⋯
L(s)  = 1  + (−0.726 + 0.686i)2-s + (0.807 + 1.39i)3-s + (0.0569 − 0.998i)4-s + (0.929 − 0.368i)5-s + (−1.54 − 0.462i)6-s + (−0.878 − 0.478i)7-s + (0.644 + 0.764i)8-s + (−0.803 + 1.39i)9-s + (−0.422 + 0.906i)10-s + (0.848 + 1.46i)11-s + (1.44 − 0.726i)12-s + 0.857i·13-s + (0.966 − 0.255i)14-s + (1.26 + 1.00i)15-s + (−0.993 − 0.113i)16-s + (−0.325 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.436 - 0.899i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656434 + 1.04824i\)
\(L(\frac12)\) \(\approx\) \(0.656434 + 1.04824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 - 0.971i)T \)
5 \( 1 + (-2.07 + 0.823i)T \)
7 \( 1 + (2.32 + 1.26i)T \)
good3 \( 1 + (-1.39 - 2.42i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.81 - 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.09iT - 13T^{2} \)
17 \( 1 + (1.34 + 2.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.00896 - 0.00517i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.25 - 2.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.20iT - 29T^{2} \)
31 \( 1 + (-2.49 - 4.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.224 - 0.389i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + 9.44iT - 43T^{2} \)
47 \( 1 + (-1.23 - 0.715i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.17 + 3.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.320 + 0.185i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.32 - 2.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.31iT - 71T^{2} \)
73 \( 1 + (4.50 + 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.69 - 2.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (-0.909 - 0.525i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.06T + 97T^{2} \)
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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

−12.05059871287908809740500421065, −10.49368232192883254031230060100, −9.916232346410860051145859217142, −9.309343631482604110597789083432, −8.884444351607076792215740820260, −7.30909718115954768133407203848, −6.36450271799000196882056297755, −4.94992145376162060751614258660, −4.05801669719167243774291396221, −2.13728836378814808783056904925, 1.22966401049228604252464830675, 2.64078586376956855928074200856, 3.31132211707547849002644497397, 6.11584576968384246076527387837, 6.63937557202133512844238357208, 7.954376679879038425712396433216, 8.737143698685809392393712973543, 9.435935631094760615559270527754, 10.56097134788343587908332354214, 11.61165247044323923271746165506

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