Properties

Label 2-280-35.12-c1-0-0
Degree $2$
Conductor $280$
Sign $-0.904 + 0.426i$
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  + (−0.739 + 2.75i)3-s + (−2.23 + 0.0819i)5-s + (−1.82 − 1.91i)7-s + (−4.46 − 2.57i)9-s + (2.37 + 4.11i)11-s + (−1.92 − 1.92i)13-s + (1.42 − 6.22i)15-s + (−6.59 − 1.76i)17-s + (−0.0439 + 0.0761i)19-s + (6.62 − 3.63i)21-s + (0.0650 + 0.242i)23-s + (4.98 − 0.366i)25-s + (4.35 − 4.35i)27-s + 0.284i·29-s + (−3.69 + 2.13i)31-s + ⋯
L(s)  = 1  + (−0.426 + 1.59i)3-s + (−0.999 + 0.0366i)5-s + (−0.691 − 0.722i)7-s + (−1.48 − 0.859i)9-s + (0.715 + 1.23i)11-s + (−0.534 − 0.534i)13-s + (0.368 − 1.60i)15-s + (−1.59 − 0.428i)17-s + (−0.0100 + 0.0174i)19-s + (1.44 − 0.792i)21-s + (0.0135 + 0.0505i)23-s + (0.997 − 0.0732i)25-s + (0.838 − 0.838i)27-s + 0.0529i·29-s + (−0.662 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.904 + 0.426i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0687589 - 0.306712i\)
\(L(\frac12)\) \(\approx\) \(0.0687589 - 0.306712i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0819i)T \)
7 \( 1 + (1.82 + 1.91i)T \)
good3 \( 1 + (0.739 - 2.75i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.37 - 4.11i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.92 + 1.92i)T + 13iT^{2} \)
17 \( 1 + (6.59 + 1.76i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.0439 - 0.0761i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0650 - 0.242i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.284iT - 29T^{2} \)
31 \( 1 + (3.69 - 2.13i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.93 - 1.05i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 9.16iT - 41T^{2} \)
43 \( 1 + (6.72 - 6.72i)T - 43iT^{2} \)
47 \( 1 + (-1.80 - 6.75i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.50 - 2.27i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.56 - 2.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.84 + 2.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.588 + 2.19i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.53T + 71T^{2} \)
73 \( 1 + (-2.33 + 8.72i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (10.7 + 6.18i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 + (-7.02 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.72 + 5.72i)T - 97iT^{2} \)
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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

−12.13609530831698898699299870429, −11.26561534866061646013075719513, −10.49217172021071798225743983223, −9.697322283912047853090189526873, −8.935623423742349058710740115059, −7.44673527138592077357943987494, −6.51508951518383976833704516155, −4.78653100452569828805924920196, −4.31839973010439156182774394147, −3.23206732150229142141043247202, 0.23928260473791384981071918252, 2.19942484558186003797296480358, 3.76353852293249497816898252951, 5.54577853304734441927627521487, 6.61799632347017021944547318276, 7.10153281871063263550871095428, 8.461604644124972112325329602072, 8.940505024583194961980902674774, 10.80984278677231755185608036249, 11.74472937475562028720276248112

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