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Properties

Label 2-280-280.69-c2-0-12
Degree $2$
Conductor $280$
Sign $-0.225 - 0.974i$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 5.74i·3-s − 4·4-s + (1.12 + 4.87i)5-s + 11.4·6-s + 7i·7-s − 8i·8-s − 23.9·9-s + (−9.74 + 2.25i)10-s + 22.9i·12-s + 14.2i·13-s − 14·14-s + (27.9 − 6.48i)15-s + 16·16-s − 47.9i·18-s + 20.1·19-s + ⋯
L(s)  = 1  + i·2-s − 1.91i·3-s − 4-s + (0.225 + 0.974i)5-s + 1.91·6-s + i·7-s − i·8-s − 2.66·9-s + (−0.974 + 0.225i)10-s + 1.91i·12-s + 1.09i·13-s − 14-s + (1.86 − 0.432i)15-s + 16-s − 2.66i·18-s + 1.06·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.225 - 0.974i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ -0.225 - 0.974i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.671969 + 0.845567i\)
\(L(\frac12)\) \(\approx\) \(0.671969 + 0.845567i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-1.12 - 4.87i)T \)
7 \( 1 - 7iT \)
good3 \( 1 + 5.74iT - 9T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 - 14.2iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 - 20.1T + 361T^{2} \)
23 \( 1 - 44.8iT - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 117.T + 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 + 89.7T + 5.04e3T^{2} \)
73 \( 1 + 5.32e3T^{2} \)
79 \( 1 + 89.7T + 6.24e3T^{2} \)
83 \( 1 + 97.9iT - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 9.40e3T^{2} \)
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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

−11.94564114423236827463251697924, −11.46094076957461694500493175100, −9.605391721566712319733195790594, −8.726183713891372052337522386786, −7.59278272379683870839738427636, −7.09187283000678463428008831397, −6.14746410677005567460043785525, −5.50521714600397236987063425049, −3.15450669190579261230909154855, −1.70218016272876016319765340176, 0.55705475856640411378650876433, 2.97843694399853507727034249702, 4.08383353022252602641706966065, 4.79045247384708943620255980385, 5.64897067882349695251305370357, 8.104946585605901743523054974223, 8.873548119841325421064769264047, 9.839823237528745543520505384353, 10.29266692907374024827155824083, 11.04506576179332765814292343863

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