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Properties

Label 2-280-280.69-c2-0-7
Degree $2$
Conductor $280$
Sign $-0.727 + 0.686i$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.65 + 1.12i)2-s + 2.74i·3-s + (1.47 − 3.71i)4-s + (−4.72 + 1.63i)5-s + (−3.09 − 4.54i)6-s + (1.41 + 6.85i)7-s + (1.75 + 7.80i)8-s + 1.43·9-s + (5.98 − 8.01i)10-s + 14.1i·11-s + (10.2 + 4.04i)12-s − 5.89i·13-s + (−10.0 − 9.75i)14-s + (−4.48 − 12.9i)15-s + (−11.6 − 10.9i)16-s − 10.0·17-s + ⋯
L(s)  = 1  + (−0.826 + 0.562i)2-s + 0.916i·3-s + (0.367 − 0.929i)4-s + (−0.945 + 0.326i)5-s + (−0.515 − 0.757i)6-s + (0.201 + 0.979i)7-s + (0.218 + 0.975i)8-s + 0.159·9-s + (0.598 − 0.801i)10-s + 1.29i·11-s + (0.852 + 0.336i)12-s − 0.453i·13-s + (−0.717 − 0.696i)14-s + (−0.299 − 0.866i)15-s + (−0.729 − 0.683i)16-s − 0.588·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.727 + 0.686i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ -0.727 + 0.686i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.187212 - 0.471310i\)
\(L(\frac12)\) \(\approx\) \(0.187212 - 0.471310i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.65 - 1.12i)T \)
5 \( 1 + (4.72 - 1.63i)T \)
7 \( 1 + (-1.41 - 6.85i)T \)
good3 \( 1 - 2.74iT - 9T^{2} \)
11 \( 1 - 14.1iT - 121T^{2} \)
13 \( 1 + 5.89iT - 169T^{2} \)
17 \( 1 + 10.0T + 289T^{2} \)
19 \( 1 + 18.9T + 361T^{2} \)
23 \( 1 - 11.5iT - 529T^{2} \)
29 \( 1 + 31.7iT - 841T^{2} \)
31 \( 1 + 48.2iT - 961T^{2} \)
37 \( 1 + 39.7T + 1.36e3T^{2} \)
41 \( 1 - 15.8iT - 1.68e3T^{2} \)
43 \( 1 - 30.0T + 1.84e3T^{2} \)
47 \( 1 + 83.3T + 2.20e3T^{2} \)
53 \( 1 - 17.9T + 2.80e3T^{2} \)
59 \( 1 - 109.T + 3.48e3T^{2} \)
61 \( 1 + 35.4T + 3.72e3T^{2} \)
67 \( 1 - 44.5T + 4.48e3T^{2} \)
71 \( 1 + 75.5T + 5.04e3T^{2} \)
73 \( 1 + 81.9T + 5.32e3T^{2} \)
79 \( 1 - 89.6T + 6.24e3T^{2} \)
83 \( 1 - 107. iT - 6.88e3T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + 13.4T + 9.40e3T^{2} \)
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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

−11.86577184218238409178030079446, −11.07116198922814126524129143242, −10.12989048641848954918846015687, −9.418993271224747936479604742280, −8.407947234115547249958387501489, −7.54321261136711130587151127771, −6.47098627857595332246512523553, −5.10707034289177475355329392759, −4.14973644414490022680950936875, −2.25777907559204430000321614599, 0.33939681536176515055382532436, 1.54066002522956689377994588951, 3.36888316147188837446095707451, 4.46415604437187791410830246982, 6.67656949858128967675575307079, 7.20985436938536264360133711637, 8.331651643823089703713259868091, 8.756646667354633634021572073649, 10.40320796097958237702882406299, 11.02450496461169181348984619788

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