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Properties

Label 2-280-7.4-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.266 - 0.963i$
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  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−2.5 + 0.866i)7-s + (1 + 1.73i)9-s + (−1 + 1.73i)11-s − 0.999·15-s + (−2 + 3.46i)17-s + (1 + 1.73i)19-s + (0.500 − 2.59i)21-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s − 5·27-s + 9·29-s + (−2 + 3.46i)31-s + (−0.999 − 1.73i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.944 + 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.301 + 0.522i)11-s − 0.258·15-s + (−0.485 + 0.840i)17-s + (0.229 + 0.397i)19-s + (0.109 − 0.566i)21-s + (−0.104 − 0.180i)23-s + (−0.0999 + 0.173i)25-s − 0.962·27-s + 1.67·29-s + (−0.359 + 0.622i)31-s + (−0.174 − 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591213 + 0.777139i\)
\(L(\frac12)\) \(\approx\) \(0.591213 + 0.777139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 + 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11T + 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−12.24422165925907341851244897416, −10.92202508336561741962648213581, −10.27163361767837742736937662996, −9.567019877068292669335617643125, −8.361505793328924693910925457053, −7.11272494485200689354641110841, −6.15057758984233623909254845878, −5.04160543654805963469012000655, −3.78086943524880099446214105900, −2.30683945843488954610806465251, 0.77596960107209571803117472311, 2.84302711075337274392253178577, 4.28899712693718974078982480025, 5.72469742834559554874646809618, 6.62122539626389563929799094781, 7.48073461747122807341826914782, 8.865511007557193034276711723516, 9.630427545704760159689836614916, 10.62121449087915718973300606748, 11.79460291339202189941915277972

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