Properties

Label 2-280-56.3-c1-0-22
Degree $2$
Conductor $280$
Sign $0.827 - 0.561i$
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  + (1.37 + 0.336i)2-s + (1.75 + 1.01i)3-s + (1.77 + 0.925i)4-s + (−0.5 − 0.866i)5-s + (2.06 + 1.98i)6-s + (−1.63 − 2.07i)7-s + (2.12 + 1.86i)8-s + (0.552 + 0.957i)9-s + (−0.394 − 1.35i)10-s + (−0.572 + 0.991i)11-s + (2.17 + 3.42i)12-s − 0.714·13-s + (−1.55 − 3.40i)14-s − 2.02i·15-s + (2.28 + 3.28i)16-s + (−1.98 − 1.14i)17-s + ⋯
L(s)  = 1  + (0.971 + 0.238i)2-s + (1.01 + 0.584i)3-s + (0.886 + 0.462i)4-s + (−0.223 − 0.387i)5-s + (0.844 + 0.809i)6-s + (−0.619 − 0.785i)7-s + (0.750 + 0.660i)8-s + (0.184 + 0.319i)9-s + (−0.124 − 0.429i)10-s + (−0.172 + 0.299i)11-s + (0.627 + 0.987i)12-s − 0.198·13-s + (−0.414 − 0.910i)14-s − 0.523i·15-s + (0.571 + 0.820i)16-s + (−0.480 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.827 - 0.561i$
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: \((2,\ 280,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.54442 + 0.782298i\)
\(L(\frac12)\) \(\approx\) \(2.54442 + 0.782298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.336i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.63 + 2.07i)T \)
good3 \( 1 + (-1.75 - 1.01i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.572 - 0.991i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.714T + 13T^{2} \)
17 \( 1 + (1.98 + 1.14i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.36 - 1.94i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.00 - 2.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.54iT - 29T^{2} \)
31 \( 1 + (0.590 - 1.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.72 + 3.30i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + (-2.60 - 4.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.25 - 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.5 - 6.07i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.13 + 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.08 + 7.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 + (4.71 + 2.72i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.20 - 4.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.78iT - 83T^{2} \)
89 \( 1 + (11.0 - 6.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.18iT - 97T^{2} \)
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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

−12.30217524390926230386748067185, −11.02344102269091129394323763299, −10.06656521116590654082188202792, −9.050780527909616074189059030546, −7.964840777303883597882855476573, −7.08557300810220557917067499898, −5.81882552991286785910782197756, −4.32285453708180814588510066801, −3.79562641495840391406518863417, −2.47500164934060602156504032633, 2.25863658596009434699407873201, 2.94195860287825898168219460487, 4.27162375058057279323970547400, 5.84291278987608569795615618383, 6.70556376547476429967277219605, 7.80368865850929674752492275817, 8.778110666528290233851400049873, 9.975040323007251759318621460094, 11.06828115688559334733772788456, 11.99957678374107560090405566739

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