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Properties

Label 2-280-7.4-c3-0-8
Degree $2$
Conductor $280$
Sign $0.335 - 0.942i$
Analytic cond. $16.5205$
Root an. cond. $4.06454$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 − 2.14i)3-s + (2.5 + 4.33i)5-s + (17.8 + 5.09i)7-s + (10.4 + 18.0i)9-s + (−26.0 + 45.0i)11-s − 53.4·13-s + 12.3·15-s + (−2.44 + 4.23i)17-s + (−51.1 − 88.5i)19-s + (32.9 − 31.8i)21-s + (105. + 183. i)23-s + (−12.5 + 21.6i)25-s + 118.·27-s + 0.173·29-s + (−123. + 214. i)31-s + ⋯
L(s)  = 1  + (0.238 − 0.412i)3-s + (0.223 + 0.387i)5-s + (0.961 + 0.275i)7-s + (0.386 + 0.669i)9-s + (−0.712 + 1.23i)11-s − 1.13·13-s + 0.213·15-s + (−0.0349 + 0.0604i)17-s + (−0.617 − 1.06i)19-s + (0.342 − 0.331i)21-s + (0.960 + 1.66i)23-s + (−0.100 + 0.173i)25-s + 0.844·27-s + 0.00110·29-s + (−0.716 + 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.335 - 0.942i$
Analytic conductor: \(16.5205\)
Root analytic conductor: \(4.06454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :3/2),\ 0.335 - 0.942i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.910321278\)
\(L(\frac12)\) \(\approx\) \(1.910321278\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-17.8 - 5.09i)T \)
good3 \( 1 + (-1.23 + 2.14i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (26.0 - 45.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 53.4T + 2.19e3T^{2} \)
17 \( 1 + (2.44 - 4.23i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (51.1 + 88.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-105. - 183. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 0.173T + 2.43e4T^{2} \)
31 \( 1 + (123. - 214. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-41.5 - 71.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 210.T + 6.89e4T^{2} \)
43 \( 1 - 89.9T + 7.95e4T^{2} \)
47 \( 1 + (18.8 + 32.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-356. + 618. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (119. - 206. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-130. - 226. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (423. - 733. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 686.T + 3.57e5T^{2} \)
73 \( 1 + (-155. + 269. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (254. + 440. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 633.T + 5.71e5T^{2} \)
89 \( 1 + (479. + 829. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 962.T + 9.12e5T^{2} \)
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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

−11.57121845565739190350867515996, −10.66680205100925353795517249525, −9.790086460421892904833060610456, −8.655520748352436388146515985180, −7.31406671718290455432732494897, −7.26600110996367685285918340619, −5.30232263782197121507469567077, −4.65094116421294233764464987949, −2.62529506315183127638918697631, −1.75782551795271121432517834976, 0.70723576501709867875047204968, 2.46774947836751756196771241530, 4.00796900132496428401062543852, 4.97120053413469239288410663248, 6.11088273551592193558113754276, 7.51962968562301655086111691644, 8.416891860037571849467406987237, 9.281239458006898871817407887919, 10.36839545615632263982904661258, 11.04022889591232232689563012031

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