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Properties

Label 2-280-280.69-c2-0-55
Degree $2$
Conductor $280$
Sign $0.486 + 0.873i$
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  + (−1.99 + 0.178i)2-s + 5.14i·3-s + (3.93 − 0.709i)4-s + (−3.67 + 3.39i)5-s + (−0.915 − 10.2i)6-s + (3.38 − 6.12i)7-s + (−7.71 + 2.11i)8-s − 17.4·9-s + (6.71 − 7.41i)10-s − 18.5i·11-s + (3.64 + 20.2i)12-s − 12.0i·13-s + (−5.65 + 12.8i)14-s + (−17.4 − 18.8i)15-s + (14.9 − 5.58i)16-s − 19.5·17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0890i)2-s + 1.71i·3-s + (0.984 − 0.177i)4-s + (−0.734 + 0.678i)5-s + (−0.152 − 1.70i)6-s + (0.483 − 0.875i)7-s + (−0.964 + 0.264i)8-s − 1.93·9-s + (0.671 − 0.741i)10-s − 1.68i·11-s + (0.304 + 1.68i)12-s − 0.925i·13-s + (−0.404 + 0.914i)14-s + (−1.16 − 1.25i)15-s + (0.937 − 0.349i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.486 + 0.873i$
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: \((2,\ 280,\ (\ :1),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.269520 - 0.158369i\)
\(L(\frac12)\) \(\approx\) \(0.269520 - 0.158369i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.178i)T \)
5 \( 1 + (3.67 - 3.39i)T \)
7 \( 1 + (-3.38 + 6.12i)T \)
good3 \( 1 - 5.14iT - 9T^{2} \)
11 \( 1 + 18.5iT - 121T^{2} \)
13 \( 1 + 12.0iT - 169T^{2} \)
17 \( 1 + 19.5T + 289T^{2} \)
19 \( 1 + 6.87T + 361T^{2} \)
23 \( 1 - 20.3iT - 529T^{2} \)
29 \( 1 + 2.31iT - 841T^{2} \)
31 \( 1 - 6.49iT - 961T^{2} \)
37 \( 1 - 30.2T + 1.36e3T^{2} \)
41 \( 1 + 48.2iT - 1.68e3T^{2} \)
43 \( 1 + 70.0T + 1.84e3T^{2} \)
47 \( 1 + 68.9T + 2.20e3T^{2} \)
53 \( 1 - 14.6T + 2.80e3T^{2} \)
59 \( 1 + 23.3T + 3.48e3T^{2} \)
61 \( 1 - 65.4T + 3.72e3T^{2} \)
67 \( 1 + 38.4T + 4.48e3T^{2} \)
71 \( 1 - 66.3T + 5.04e3T^{2} \)
73 \( 1 - 49.3T + 5.32e3T^{2} \)
79 \( 1 + 44.8T + 6.24e3T^{2} \)
83 \( 1 + 30.7iT - 6.88e3T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 + 36.8T + 9.40e3T^{2} \)
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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

−10.99577942493859836471731183728, −10.66994335403272060957947797736, −9.750736950519819663421743747294, −8.565150153574051151671424328372, −8.029554383258335613234457493476, −6.64037340948876560215590699402, −5.35976082220815871481258615653, −3.89181096320284471338117735344, −3.06953161824988348723375271061, −0.21021154465022058441143635115, 1.58072641177013094171926554437, 2.34480387612418787088107187659, 4.70301009376629618360466265480, 6.41858809751537995138155431988, 7.05612654280594306386954867467, 8.055124665114984329681513971174, 8.600291959798711744161948137100, 9.574276364188088922729730949861, 11.31191014679078188018769195484, 11.77238256608189651891675487228

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