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Properties

Label 2-280-280.19-c1-0-0
Degree $2$
Conductor $280$
Sign $0.746 - 0.665i$
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  + (−0.498 − 1.32i)2-s + (−0.929 − 1.60i)3-s + (−1.50 + 1.31i)4-s + (−2.18 + 0.457i)5-s + (−1.66 + 2.03i)6-s + (−1.10 + 2.40i)7-s + (2.49 + 1.33i)8-s + (−0.226 + 0.392i)9-s + (1.69 + 2.66i)10-s + (1.61 + 2.78i)11-s + (3.51 + 1.19i)12-s − 0.668i·13-s + (3.73 + 0.266i)14-s + (2.77 + 3.09i)15-s + (0.521 − 3.96i)16-s + (1.62 + 2.81i)17-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)2-s + (−0.536 − 0.929i)3-s + (−0.751 + 0.659i)4-s + (−0.978 + 0.204i)5-s + (−0.680 + 0.829i)6-s + (−0.418 + 0.908i)7-s + (0.881 + 0.471i)8-s + (−0.0755 + 0.130i)9-s + (0.536 + 0.844i)10-s + (0.485 + 0.841i)11-s + (1.01 + 0.344i)12-s − 0.185i·13-s + (0.997 + 0.0712i)14-s + (0.715 + 0.799i)15-s + (0.130 − 0.991i)16-s + (0.393 + 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320001 + 0.121915i\)
\(L(\frac12)\) \(\approx\) \(0.320001 + 0.121915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.498 + 1.32i)T \)
5 \( 1 + (2.18 - 0.457i)T \)
7 \( 1 + (1.10 - 2.40i)T \)
good3 \( 1 + (0.929 + 1.60i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.61 - 2.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.668iT - 13T^{2} \)
17 \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.77 + 3.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.41 - 7.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.17iT - 29T^{2} \)
31 \( 1 + (-2.58 - 4.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.15 + 2.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.34iT - 41T^{2} \)
43 \( 1 - 6.84iT - 43T^{2} \)
47 \( 1 + (3.12 + 1.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0130 + 0.0225i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0160 + 0.00925i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.897 - 1.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.14 - 2.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (6.30 + 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.5 + 7.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.60T + 83T^{2} \)
89 \( 1 + (1.31 + 0.760i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−11.89955139959327163508656104437, −11.42111618698367575943718004447, −10.21298188116140917010892168674, −9.167075432982797560390209590923, −8.151330177673719262889422654812, −7.22027321279749921307562530234, −6.14729422712523439079275747510, −4.53125382029477645992946423031, −3.24814217198196724397023987833, −1.67991682637248572242408928352, 0.32092589497083026338228785097, 3.98347488017951639853487621402, 4.37820650018131323000799236512, 5.79758035290578189949275190485, 6.80945731101769239453377236298, 7.920110004446280467616031443937, 8.738916518379022766674346593564, 9.978178899835519691389447858241, 10.54140488203472816772635498429, 11.48464559631464244190635834033

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