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Properties

Label 2-280-280.59-c1-0-20
Degree $2$
Conductor $280$
Sign $0.942 - 0.333i$
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  + (−1.09 + 0.889i)2-s + (0.360 − 0.624i)3-s + (0.416 − 1.95i)4-s + (1.56 + 1.59i)5-s + (0.159 + 1.00i)6-s + (−1.35 − 2.27i)7-s + (1.28 + 2.52i)8-s + (1.24 + 2.14i)9-s + (−3.14 − 0.363i)10-s + (1.95 − 3.38i)11-s + (−1.07 − 0.964i)12-s + 2.55i·13-s + (3.51 + 1.28i)14-s + (1.56 − 0.400i)15-s + (−3.65 − 1.62i)16-s + (2.55 − 4.43i)17-s + ⋯
L(s)  = 1  + (−0.777 + 0.629i)2-s + (0.208 − 0.360i)3-s + (0.208 − 0.978i)4-s + (0.699 + 0.714i)5-s + (0.0650 + 0.410i)6-s + (−0.512 − 0.858i)7-s + (0.453 + 0.891i)8-s + (0.413 + 0.716i)9-s + (−0.993 − 0.115i)10-s + (0.589 − 1.02i)11-s + (−0.309 − 0.278i)12-s + 0.708i·13-s + (0.938 + 0.344i)14-s + (0.402 − 0.103i)15-s + (−0.913 − 0.406i)16-s + (0.620 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07092 + 0.183914i\)
\(L(\frac12)\) \(\approx\) \(1.07092 + 0.183914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 - 0.889i)T \)
5 \( 1 + (-1.56 - 1.59i)T \)
7 \( 1 + (1.35 + 2.27i)T \)
good3 \( 1 + (-0.360 + 0.624i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.95 + 3.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.55iT - 13T^{2} \)
17 \( 1 + (-2.55 + 4.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.25 + 1.88i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.50 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.39iT - 29T^{2} \)
31 \( 1 + (1.59 - 2.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.850 + 1.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.97iT - 41T^{2} \)
43 \( 1 - 0.898iT - 43T^{2} \)
47 \( 1 + (5.64 - 3.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.92 + 5.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.2 + 5.91i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.94 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.68 + 1.54i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + (2.59 - 4.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.84 - 5.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.786T + 83T^{2} \)
89 \( 1 + (0.679 - 0.392i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−11.48211667763815929918540835462, −10.84284219505457111699591936803, −9.739929275038289886220401989062, −9.268985260475641130931038961893, −7.77037754936451615904514353494, −7.09802708267714198822915720766, −6.33691146688171870971515916112, −5.08757248999192087597927496516, −3.12923732132745406697402822626, −1.39072824633516242869883597202, 1.46613584055094114157111167659, 3.00411783306091269893514311151, 4.31707377578673263649565953923, 5.82960416525030436476045336322, 7.02218589753284271070792174537, 8.449061651699418401176084592100, 9.128280040013936159013262184124, 9.846362455578322275594172648233, 10.44379886857238578882963591324, 12.15539547838931082397016765765

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