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Properties

Label 2-1120-35.4-c1-0-36
Degree $2$
Conductor $1120$
Sign $0.977 - 0.211i$
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 + (0.946 − 2.02i)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 + (−3.20 − 3.83i)25-s + 1.43i·27-s − 4·29-s + (−1.83 + 3.17i)31-s + ⋯
L(s)  = 1  + (1.28 + 0.739i)3-s + (0.423 − 0.905i)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 + (−0.641 − 0.767i)25-s + 0.276i·27-s − 0.742·29-s + (−0.329 + 0.570i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (1089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.987668906\)
\(L(\frac12)\) \(\approx\) \(2.987668906\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.946 + 2.02i)T \)
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

−9.326299776806612218136826699568, −9.072549882555976028522233707561, −8.547285855862963344539059302334, −7.84774670690223532243270931088, −6.30722715701138731402056630724, −5.52855335225141802646439040638, −4.45075214642651420709520414723, −3.71320336020537760737244515000, −2.57938161335895602326850201175, −1.44183811041855788038249279015, 1.61204821292400320678490386824, 2.20984586785649966844126826484, 3.48415203113116971072046379561, 4.24447586929055389788678157277, 5.72917590582598443126563177872, 6.82610126806680644253539079159, 7.47660854867883730600797337710, 7.86151598528471964500348398330, 8.917871071955311590902634624867, 9.963322479254411270488391576776

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