Properties

Label 2-280-35.12-c1-0-4
Degree $2$
Conductor $280$
Sign $0.727 - 0.686i$
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  + (0.0752 − 0.280i)3-s + (1.39 + 1.74i)5-s + (−2.39 + 1.13i)7-s + (2.52 + 1.45i)9-s + (1.58 + 2.74i)11-s + (−1.12 − 1.12i)13-s + (0.595 − 0.259i)15-s + (2.91 + 0.781i)17-s + (1.03 − 1.79i)19-s + (0.137 + 0.756i)21-s + (−0.593 − 2.21i)23-s + (−1.11 + 4.87i)25-s + (1.21 − 1.21i)27-s − 1.39i·29-s + (0.467 − 0.269i)31-s + ⋯
L(s)  = 1  + (0.0434 − 0.162i)3-s + (0.623 + 0.782i)5-s + (−0.904 + 0.427i)7-s + (0.841 + 0.485i)9-s + (0.477 + 0.826i)11-s + (−0.312 − 0.312i)13-s + (0.153 − 0.0670i)15-s + (0.706 + 0.189i)17-s + (0.237 − 0.411i)19-s + (0.0300 + 0.165i)21-s + (−0.123 − 0.461i)23-s + (−0.223 + 0.974i)25-s + (0.233 − 0.233i)27-s − 0.259i·29-s + (0.0839 − 0.0484i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28198 + 0.509067i\)
\(L(\frac12)\) \(\approx\) \(1.28198 + 0.509067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.39 - 1.74i)T \)
7 \( 1 + (2.39 - 1.13i)T \)
good3 \( 1 + (-0.0752 + 0.280i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.58 - 2.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.12 + 1.12i)T + 13iT^{2} \)
17 \( 1 + (-2.91 - 0.781i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.03 + 1.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.593 + 2.21i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.39iT - 29T^{2} \)
31 \( 1 + (-0.467 + 0.269i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.45 + 1.72i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 + (6.50 - 6.50i)T - 43iT^{2} \)
47 \( 1 + (3.32 + 12.4i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (10.8 + 2.89i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.55 + 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.2 + 7.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.826 + 3.08i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.61T + 71T^{2} \)
73 \( 1 + (-0.827 + 3.08i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-14.7 - 8.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.62 + 4.62i)T + 83iT^{2} \)
89 \( 1 + (-8.86 + 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.71 - 3.71i)T - 97iT^{2} \)
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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

−12.14665048497673365719488876967, −10.89533895208824031865781961500, −9.806985467733141992532204940075, −9.617779932473873133859714734304, −7.946940883163353543903759205448, −6.93210095189247911617237285112, −6.21618465625434985823001831238, −4.86865643965770559140676542578, −3.29548131724102509718554699093, −2.01707651843257961711376619246, 1.21380252372267350998611209591, 3.29376436647840666553193901687, 4.43832045000773047746384256588, 5.79569552003520624340691101590, 6.67274597453174324227260975474, 7.900730264669841009429293827795, 9.292263358602800873566059255912, 9.596922266554420471981657809453, 10.60407511477713022480940371663, 11.98530630929214857681883937259

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