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Properties

Label 2-280-35.4-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.741 - 0.671i$
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  + (−0.277 − 0.159i)3-s + (−2.00 + 0.986i)5-s + (−1.50 + 2.17i)7-s + (−1.44 − 2.50i)9-s + (−2.08 + 3.60i)11-s + 2.89i·13-s + (0.713 + 0.0477i)15-s + (−3.72 − 2.15i)17-s + (0.979 + 1.69i)19-s + (0.765 − 0.361i)21-s + (−2.23 + 1.28i)23-s + (3.05 − 3.95i)25-s + 1.88i·27-s + 5.96·29-s + (−4.71 + 8.16i)31-s + ⋯
L(s)  = 1  + (−0.159 − 0.0923i)3-s + (−0.897 + 0.441i)5-s + (−0.569 + 0.821i)7-s + (−0.482 − 0.836i)9-s + (−0.628 + 1.08i)11-s + 0.803i·13-s + (0.184 + 0.0123i)15-s + (−0.903 − 0.521i)17-s + (0.224 + 0.389i)19-s + (0.167 − 0.0788i)21-s + (−0.465 + 0.268i)23-s + (0.610 − 0.791i)25-s + 0.363i·27-s + 1.10·29-s + (−0.846 + 1.46i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.741 - 0.671i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.741 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169897 + 0.440505i\)
\(L(\frac12)\) \(\approx\) \(0.169897 + 0.440505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 - 0.986i)T \)
7 \( 1 + (1.50 - 2.17i)T \)
good3 \( 1 + (0.277 + 0.159i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.08 - 3.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.89iT - 13T^{2} \)
17 \( 1 + (3.72 + 2.15i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.979 - 1.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 - 1.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
31 \( 1 + (4.71 - 8.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.48 + 3.16i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.19T + 41T^{2} \)
43 \( 1 + 4.73iT - 43T^{2} \)
47 \( 1 + (-3.84 + 2.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.37 + 3.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.61 - 2.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.94 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 - 5.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.58T + 71T^{2} \)
73 \( 1 + (-8.57 - 4.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.75iT - 83T^{2} \)
89 \( 1 + (1.50 + 2.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.414iT - 97T^{2} \)
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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

−12.07143196997359371172747823104, −11.57041169813353731930653077073, −10.35662093745150783810302998472, −9.317684376300331851485196684470, −8.468810125470212723794722701247, −7.14746488791561322501684429832, −6.50130968939010475112828928932, −5.11472205564098083748914593630, −3.77392953597808971468564188115, −2.50292225967117485355995349301, 0.34692909759281402855692094582, 2.93998809196108240377380609422, 4.17338483353031467656898479446, 5.31091062669421568308938103084, 6.52337103741437401753411653837, 7.949402151688553033514634913536, 8.246974440059146420461032186910, 9.684435496036197693820133893797, 10.98622967961814520506326106025, 11.06782591635871569543057648044

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