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Properties

Label 2-1120-35.9-c1-0-44
Degree $2$
Conductor $1120$
Sign $-0.534 + 0.845i$
Analytic cond. $8.94324$
Root an. cond. $2.99052$
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  + (2.21 − 1.28i)3-s + (−2.22 + 0.192i)5-s + (1.73 − 2i)7-s + (1.78 − 3.08i)9-s + (−3.26 − 5.65i)11-s + 2.86i·13-s + (−4.69 + 3.28i)15-s + (−3.17 + 1.83i)17-s + (3.26 − 5.65i)19-s + (1.28 − 6.65i)21-s + (−3.57 − 2.06i)23-s + (4.92 − 0.858i)25-s − 1.43i·27-s − 4·29-s + (1.83 + 3.17i)31-s + ⋯
L(s)  = 1  + (1.28 − 0.739i)3-s + (−0.996 + 0.0861i)5-s + (0.654 − 0.755i)7-s + (0.593 − 1.02i)9-s + (−0.984 − 1.70i)11-s + 0.793i·13-s + (−1.21 + 0.847i)15-s + (−0.769 + 0.444i)17-s + (0.748 − 1.29i)19-s + (0.279 − 1.45i)21-s + (−0.744 − 0.429i)23-s + (0.985 − 0.171i)25-s − 0.276i·27-s − 0.742·29-s + (0.329 + 0.570i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.534 + 0.845i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.534 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.802740622\)
\(L(\frac12)\) \(\approx\) \(1.802740622\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.192i)T \)
7 \( 1 + (-1.73 + 2i)T \)
good3 \( 1 + (-2.21 + 1.28i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.26 + 5.65i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.86iT - 13T^{2} \)
17 \( 1 + (3.17 - 1.83i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.26 + 5.65i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.57 + 2.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-1.83 - 3.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.695 + 0.401i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.56T + 41T^{2} \)
43 \( 1 + 8.24iT - 43T^{2} \)
47 \( 1 + (-3.57 - 2.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.695 - 0.401i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.02 - 1.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.84 + 8.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.95 - 2.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.72T + 71T^{2} \)
73 \( 1 + (-8.13 + 4.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.83 + 3.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.246iT - 83T^{2} \)
89 \( 1 + (-6.84 + 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.7iT - 97T^{2} \)
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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

−9.105868391795326482906612452975, −8.548862501837235481603697676114, −7.952834806694765701964469088409, −7.34537960041076429892179620459, −6.56062356532347740095367500094, −5.09665928681366324984954205806, −3.99209706593556475377179652117, −3.21632884492739267239648913244, −2.19095357112049990579527375330, −0.65966080473665998450645072004, 2.03770674961983949942712967257, 2.90763565029322141733629795216, 3.97912362524072295731687417579, 4.69899159240809157269605395766, 5.56508721344521932357286336392, 7.26796872700155470318123803445, 7.967716203123789690043720028489, 8.223374084392790920601895477679, 9.372473645119536051083144703197, 9.854874465783253823650596670576

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