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Properties

Label 2-280-56.3-c1-0-9
Degree $2$
Conductor $280$
Sign $-0.553 + 0.832i$
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  + (−1.22 − 0.707i)2-s + (−2.72 − 1.57i)3-s + (0.999 + 1.73i)4-s + (0.5 + 0.866i)5-s + (2.22 + 3.85i)6-s + (2.5 − 0.866i)7-s − 2.82i·8-s + (3.44 + 5.97i)9-s − 1.41i·10-s + (2.44 − 4.24i)11-s − 6.29i·12-s − 0.449·13-s + (−3.67 − 0.707i)14-s − 3.14i·15-s + (−2.00 + 3.46i)16-s + (−1.77 − 1.02i)17-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (−1.57 − 0.908i)3-s + (0.499 + 0.866i)4-s + (0.223 + 0.387i)5-s + (0.908 + 1.57i)6-s + (0.944 − 0.327i)7-s − 0.999i·8-s + (1.14 + 1.99i)9-s − 0.447i·10-s + (0.738 − 1.27i)11-s − 1.81i·12-s − 0.124·13-s + (−0.981 − 0.188i)14-s − 0.812i·15-s + (−0.500 + 0.866i)16-s + (−0.430 − 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251767 - 0.469850i\)
\(L(\frac12)\) \(\approx\) \(0.251767 - 0.469850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good3 \( 1 + (2.72 + 1.57i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.44 + 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 + (1.77 + 1.02i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.67 - 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.949 + 0.548i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 10.0iT - 29T^{2} \)
31 \( 1 + (-3.22 + 5.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 + 1.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.36iT - 41T^{2} \)
43 \( 1 + 4.55T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.22 - 2.43i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.12 + 5.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.17 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.17 + 2.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (2.32 + 1.34i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.674 - 0.389i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 + (0.398 - 0.230i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.02iT - 97T^{2} \)
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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

−11.44962529304273471909483521657, −10.91587047832998234563342569651, −10.01643709815444322276537021957, −8.493897877799311899413935143154, −7.62497987485950307595613875208, −6.59914667291269252852283189549, −5.87777112152146549796695598733, −4.24514754222682582703660249740, −2.08550631939329448872035821156, −0.70659462693298453991776413315, 1.51120056911548877095012953310, 4.61924441858115855013987352411, 5.03674933363583516986998518936, 6.26970709892000490355201848940, 7.05438092233608930529434958414, 8.619261602319025562529121165025, 9.430621613898929041223202094539, 10.32421903596451213898741706233, 11.04953111226226761175726710088, 11.82200951542367474190375196333

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