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Properties

Label 2-280-280.139-c1-0-29
Degree $2$
Conductor $280$
Sign $0.465 - 0.885i$
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  + (0.918 + 1.07i)2-s + 1.19·3-s + (−0.313 + 1.97i)4-s + (2.22 + 0.234i)5-s + (1.09 + 1.28i)6-s + (2.11 − 1.58i)7-s + (−2.41 + 1.47i)8-s − 1.58·9-s + (1.78 + 2.60i)10-s − 3.65·11-s + (−0.373 + 2.35i)12-s − 5.56i·13-s + (3.65 + 0.822i)14-s + (2.64 + 0.279i)15-s + (−3.80 − 1.23i)16-s − 0.808·17-s + ⋯
L(s)  = 1  + (0.649 + 0.760i)2-s + 0.687·3-s + (−0.156 + 0.987i)4-s + (0.994 + 0.105i)5-s + (0.446 + 0.522i)6-s + (0.800 − 0.599i)7-s + (−0.852 + 0.521i)8-s − 0.527·9-s + (0.565 + 0.824i)10-s − 1.10·11-s + (−0.107 + 0.678i)12-s − 1.54i·13-s + (0.975 + 0.219i)14-s + (0.683 + 0.0722i)15-s + (−0.950 − 0.309i)16-s − 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.465 - 0.885i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.465 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96236 + 1.18498i\)
\(L(\frac12)\) \(\approx\) \(1.96236 + 1.18498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.918 - 1.07i)T \)
5 \( 1 + (-2.22 - 0.234i)T \)
7 \( 1 + (-2.11 + 1.58i)T \)
good3 \( 1 - 1.19T + 3T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 + 5.56iT - 13T^{2} \)
17 \( 1 + 0.808T + 17T^{2} \)
19 \( 1 - 4.54iT - 19T^{2} \)
23 \( 1 + 1.75T + 23T^{2} \)
29 \( 1 - 8.36iT - 29T^{2} \)
31 \( 1 + 4.73T + 31T^{2} \)
37 \( 1 - 6.15T + 37T^{2} \)
41 \( 1 + 7.65iT - 41T^{2} \)
43 \( 1 + 2.66iT - 43T^{2} \)
47 \( 1 + 4.79iT - 47T^{2} \)
53 \( 1 - 2.98T + 53T^{2} \)
59 \( 1 - 8.92iT - 59T^{2} \)
61 \( 1 - 4.08T + 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 8.17iT - 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + 2.49iT - 79T^{2} \)
83 \( 1 + 5.75T + 83T^{2} \)
89 \( 1 - 4.95iT - 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−12.45886147424251506519974963275, −10.96682738886661355779358425211, −10.18381896712830813012867019615, −8.783477899686610681214982513366, −8.054147900754362462779871014784, −7.27501407795126044980089563061, −5.70992692936440441517345519532, −5.24351852569206136060500799319, −3.57319738579144397240871742830, −2.41285100968228858328816219387, 2.03539207384248062535903835962, 2.66148435892495884495161657616, 4.45120015402628611307855290195, 5.40374399070557772453991866101, 6.40300438914401024401477510480, 8.091696516395647578038405283161, 9.159358195638379504639415753082, 9.662495311972473204173063875598, 11.06073504150576349475161359208, 11.53966050455253723268909370506

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