Properties

Label 2-280-56.19-c1-0-6
Degree $2$
Conductor $280$
Sign $-0.567 - 0.823i$
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  + (1.11 + 0.868i)2-s + (−0.502 + 0.290i)3-s + (0.491 + 1.93i)4-s + (−0.5 + 0.866i)5-s + (−0.813 − 0.112i)6-s + (−2.63 + 0.253i)7-s + (−1.13 + 2.59i)8-s + (−1.33 + 2.30i)9-s + (−1.31 + 0.532i)10-s + (0.428 + 0.742i)11-s + (−0.809 − 0.832i)12-s + 2.26·13-s + (−3.15 − 2.00i)14-s − 0.580i·15-s + (−3.51 + 1.90i)16-s + (6.65 − 3.84i)17-s + ⋯
L(s)  = 1  + (0.789 + 0.614i)2-s + (−0.290 + 0.167i)3-s + (0.245 + 0.969i)4-s + (−0.223 + 0.387i)5-s + (−0.331 − 0.0460i)6-s + (−0.995 + 0.0956i)7-s + (−0.401 + 0.915i)8-s + (−0.443 + 0.768i)9-s + (−0.414 + 0.168i)10-s + (0.129 + 0.223i)11-s + (−0.233 − 0.240i)12-s + 0.627·13-s + (−0.844 − 0.535i)14-s − 0.149i·15-s + (−0.879 + 0.476i)16-s + (1.61 − 0.931i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695600 + 1.32339i\)
\(L(\frac12)\) \(\approx\) \(0.695600 + 1.32339i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 - 0.868i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.63 - 0.253i)T \)
good3 \( 1 + (0.502 - 0.290i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.428 - 0.742i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 + (-6.65 + 3.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.17 - 2.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.17 + 1.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.76iT - 29T^{2} \)
31 \( 1 + (4.53 + 7.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.77 - 2.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.780iT - 41T^{2} \)
43 \( 1 - 7.36T + 43T^{2} \)
47 \( 1 + (0.206 - 0.358i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.0 + 6.36i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.74 - 4.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.51 + 7.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.69iT - 71T^{2} \)
73 \( 1 + (9.52 - 5.49i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.53 - 2.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.58iT - 83T^{2} \)
89 \( 1 + (-5.85 - 3.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.09iT - 97T^{2} \)
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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

−12.18359335683324583130216751941, −11.54471925400442088219290366668, −10.40168168011171235077313014490, −9.333284954633434790732411182736, −7.956456916084815892019891843352, −7.24090511601816300863745940730, −5.99609323746343097185457719493, −5.34998030912805882346387230373, −3.83030939126698927775116095103, −2.85832856730984936374825640602, 0.981636164072589594981007250623, 3.15653943310334508380563867476, 3.91723335348149277820533480574, 5.62145663507355151403548882685, 6.09986450660699328452333151116, 7.40720063452981012963433922985, 8.978295153616558363090859843882, 9.778484367184732834969882897945, 10.77752406131989921658113319977, 11.90779489627540265612831816892

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