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Properties

Label 2-280-280.69-c2-0-73
Degree $2$
Conductor $280$
Sign $-0.974 + 0.225i$
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 − 1.74i·3-s − 4·4-s + (4.87 − 1.12i)5-s − 3.48·6-s − 7i·7-s + 8i·8-s + 5.96·9-s + (−2.25 − 9.74i)10-s + 6.96i·12-s − 21.7i·13-s − 14·14-s + (−1.96 − 8.48i)15-s + 16·16-s − 11.9i·18-s − 32.1·19-s + ⋯
L(s)  = 1  − i·2-s − 0.580i·3-s − 4-s + (0.974 − 0.225i)5-s − 0.580·6-s − i·7-s + i·8-s + 0.662·9-s + (−0.225 − 0.974i)10-s + 0.580i·12-s − 1.67i·13-s − 14-s + (−0.131 − 0.565i)15-s + 16-s − 0.662i·18-s − 1.69·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.974 + 0.225i$
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: \((2,\ 280,\ (\ :1),\ -0.974 + 0.225i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.184070 - 1.60908i\)
\(L(\frac12)\) \(\approx\) \(0.184070 - 1.60908i\)
\(L(2)\) 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 \( 1 + (-4.87 + 1.12i)T \)
7 \( 1 + 7iT \)
good3 \( 1 + 1.74iT - 9T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 21.7iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + 32.1T + 361T^{2} \)
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 \( 1 + 9.60T + 3.48e3T^{2} \)
61 \( 1 - 80.0T + 3.72e3T^{2} \)
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 \( 1 + 133. iT - 6.88e3T^{2} \)
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

−11.01415975873991121332924284267, −10.28063015366404595922395370274, −9.688466632619160779950697324838, −8.388380105245402626473496476356, −7.42750663496746075811170641854, −6.05536527119852521981148437088, −4.88243964759236697288290498892, −3.57392114269976444739407280183, −2.02384707449424628895067689616, −0.858832909094274798254945025081, 2.14710593620478366639738313363, 4.16442733809072998996234787985, 5.01815115087204119561256123227, 6.33411948958469444269447632467, 6.75189554850861888903195968609, 8.485800754009883607946744035657, 9.121044808070631694747501776551, 9.895777469369921122403655766622, 10.84098135025982081402487360976, 12.38204908911474846582402321496

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