Properties

Label 2-280-35.12-c1-0-11
Degree $2$
Conductor $280$
Sign $-0.711 + 0.702i$
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  + (0.856 − 3.19i)3-s + (−1.51 + 1.64i)5-s + (−0.802 − 2.52i)7-s + (−6.87 − 3.97i)9-s + (−1.05 − 1.82i)11-s + (1.20 + 1.20i)13-s + (3.97 + 6.23i)15-s + (3.17 + 0.850i)17-s + (2.36 − 4.09i)19-s + (−8.74 + 0.406i)21-s + (1.07 + 4.00i)23-s + (−0.436 − 4.98i)25-s + (−11.5 + 11.5i)27-s − 3.65i·29-s + (−1.63 + 0.946i)31-s + ⋯
L(s)  = 1  + (0.494 − 1.84i)3-s + (−0.675 + 0.737i)5-s + (−0.303 − 0.952i)7-s + (−2.29 − 1.32i)9-s + (−0.317 − 0.550i)11-s + (0.334 + 0.334i)13-s + (1.02 + 1.61i)15-s + (0.770 + 0.206i)17-s + (0.542 − 0.939i)19-s + (−1.90 + 0.0886i)21-s + (0.223 + 0.835i)23-s + (−0.0872 − 0.996i)25-s + (−2.22 + 2.22i)27-s − 0.679i·29-s + (−0.294 + 0.169i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.711 + 0.702i$
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: \((2,\ 280,\ (\ :1/2),\ -0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437801 - 1.06696i\)
\(L(\frac12)\) \(\approx\) \(0.437801 - 1.06696i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 1.64i)T \)
7 \( 1 + (0.802 + 2.52i)T \)
good3 \( 1 + (-0.856 + 3.19i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.05 + 1.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.20 - 1.20i)T + 13iT^{2} \)
17 \( 1 + (-3.17 - 0.850i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.07 - 4.00i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 + (1.63 - 0.946i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.91 + 2.65i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.826iT - 41T^{2} \)
43 \( 1 + (-4.70 + 4.70i)T - 43iT^{2} \)
47 \( 1 + (-1.04 - 3.90i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.40 - 0.645i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.15 - 2.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.44 - 0.837i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.28 + 12.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (3.37 - 12.6i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.29 + 4.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.47 - 3.47i)T + 83iT^{2} \)
89 \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.02 + 1.02i)T - 97iT^{2} \)
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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

−11.55024639209073753182763520298, −10.95333976226138042331046364577, −9.435254475695859936923170251627, −8.153549151264825320616761918965, −7.52242755660135993181820116971, −6.88523616264141312237922942908, −5.90730088626218506242979161442, −3.69939298789886291254054506065, −2.69418908078590692294170694694, −0.861798242784509726565305780668, 2.86377554257275970934460538850, 3.90950609417636629050493402559, 4.96120250545378231757311934736, 5.71668647786113159896501589516, 7.85693800645980229100886956393, 8.589814869858623662455130182518, 9.429863961169736770210567174718, 10.08778600835802001814016920045, 11.16286160981384892875068326536, 12.07755557728397327217049699000

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