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Properties

Label 2-1120-35.4-c1-0-3
Degree $2$
Conductor $1120$
Sign $-0.999 - 0.0401i$
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  + (1.35 + 0.780i)3-s + (−1.42 + 1.72i)5-s + (−1.73 − 2i)7-s + (−0.280 − 0.486i)9-s + (−0.588 + 1.01i)11-s + 5.36i·13-s + (−3.27 + 1.21i)15-s + (−3.62 − 2.09i)17-s + (0.588 + 1.01i)19-s + (−0.780 − 4.05i)21-s + (−3.57 + 2.06i)23-s + (−0.943 − 4.91i)25-s − 5.56i·27-s − 4·29-s + (−2.09 + 3.62i)31-s + ⋯
L(s)  = 1  + (0.780 + 0.450i)3-s + (−0.636 + 0.770i)5-s + (−0.654 − 0.755i)7-s + (−0.0935 − 0.162i)9-s + (−0.177 + 0.307i)11-s + 1.48i·13-s + (−0.844 + 0.314i)15-s + (−0.880 − 0.508i)17-s + (0.134 + 0.233i)19-s + (−0.170 − 0.885i)21-s + (−0.744 + 0.429i)23-s + (−0.188 − 0.982i)25-s − 1.07i·27-s − 0.742·29-s + (−0.376 + 0.651i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0401i$
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: \((2,\ 1120,\ (\ :1/2),\ -0.999 - 0.0401i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5155541539\)
\(L(\frac12)\) \(\approx\) \(0.5155541539\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 1.72i)T \)
7 \( 1 + (1.73 + 2i)T \)
good3 \( 1 + (-1.35 - 0.780i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.588 - 1.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.36iT - 13T^{2} \)
17 \( 1 + (3.62 + 2.09i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.588 - 1.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.57 - 2.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (2.09 - 3.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.27 - 4.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.56T + 41T^{2} \)
43 \( 1 - 8.24iT - 43T^{2} \)
47 \( 1 + (-3.57 + 2.06i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.27 + 4.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.46 + 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.34 + 2.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.379 - 0.219i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (5.66 + 3.27i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.09 + 3.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 + (-0.657 - 1.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.0iT - 97T^{2} \)
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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

−9.993770317285456743695811417260, −9.537408429608406578488530297977, −8.675678826236238855744208061801, −7.75113394311402240600073974266, −6.86890857647127881480283321387, −6.42864073897488226200904411377, −4.75508838396850999587256636651, −3.83219774676867301437442213823, −3.34500347161778254163031755806, −2.10282727401371529579350172672, 0.18824782465728289373148855834, 2.00950235632776213024858267428, 3.01443936267466505413503785222, 3.92500788183735382968449788176, 5.27258174743438195419808995697, 5.87980558482808447301458749888, 7.21392926617639882186090374434, 7.910612694799667767185707493952, 8.698563924748305682710937453606, 8.972532130394755356917762781796

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