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Properties

Label 2-280-56.19-c1-0-4
Degree $2$
Conductor $280$
Sign $0.975 + 0.221i$
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  + (−0.477 − 1.33i)2-s + (−1.84 + 1.06i)3-s + (−1.54 + 1.27i)4-s + (0.5 − 0.866i)5-s + (2.29 + 1.94i)6-s + (−2.17 − 1.50i)7-s + (2.42 + 1.45i)8-s + (0.759 − 1.31i)9-s + (−1.39 − 0.252i)10-s + (2.04 + 3.54i)11-s + (1.49 − 3.98i)12-s + 4.95·13-s + (−0.966 + 3.61i)14-s + 2.12i·15-s + (0.772 − 3.92i)16-s + (2.09 − 1.20i)17-s + ⋯
L(s)  = 1  + (−0.337 − 0.941i)2-s + (−1.06 + 0.613i)3-s + (−0.772 + 0.635i)4-s + (0.223 − 0.387i)5-s + (0.936 + 0.793i)6-s + (−0.822 − 0.569i)7-s + (0.858 + 0.512i)8-s + (0.253 − 0.438i)9-s + (−0.440 − 0.0798i)10-s + (0.616 + 1.06i)11-s + (0.431 − 1.14i)12-s + 1.37·13-s + (−0.258 + 0.966i)14-s + 0.548i·15-s + (0.193 − 0.981i)16-s + (0.507 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.975 + 0.221i$
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: \((2,\ 280,\ (\ :1/2),\ 0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709456 - 0.0796085i\)
\(L(\frac12)\) \(\approx\) \(0.709456 - 0.0796085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.477 + 1.33i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.17 + 1.50i)T \)
good3 \( 1 + (1.84 - 1.06i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.04 - 3.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.95T + 13T^{2} \)
17 \( 1 + (-2.09 + 1.20i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.22 - 3.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.443 + 0.255i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.08iT - 29T^{2} \)
31 \( 1 + (-2.30 - 3.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.64 - 4.99i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.81iT - 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (-0.698 + 1.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.79 - 3.92i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.82 + 5.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.33 - 2.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (-6.99 + 4.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.21 - 1.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.94iT - 83T^{2} \)
89 \( 1 + (3.81 + 2.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.67iT - 97T^{2} \)
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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

−11.74540396080241436869127088434, −10.83969228344080980459704835705, −9.946135541834408182007157043291, −9.560032499596916891552087225587, −8.225297174531135959606525847789, −6.81894314853238614437627266945, −5.54350020416876127628196508332, −4.44352920439935326937831508725, −3.42924773582807374344805610428, −1.19996821332825791721482324192, 0.912633733133135056946500130768, 3.51419904741960915676540591067, 5.42878771657357577966944952698, 6.22428605516544267404156600569, 6.49968454088718616770292602432, 7.86643219100450739128888794456, 8.981672622446999925302777144740, 9.827626921939527314803105203798, 11.10771144983368655254890084531, 11.70227100080053506384595055523

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