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Properties

Label 2-1120-40.3-c1-0-2
Degree $2$
Conductor $1120$
Sign $-0.287 - 0.957i$
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  + (−0.683 − 0.683i)3-s + (−2.03 − 0.917i)5-s + (0.707 + 0.707i)7-s − 2.06i·9-s − 5.41·11-s + (3.15 − 3.15i)13-s + (0.766 + 2.01i)15-s + (−2.46 + 2.46i)17-s + 3.87i·19-s − 0.966i·21-s + (−0.659 + 0.659i)23-s + (3.31 + 3.74i)25-s + (−3.46 + 3.46i)27-s + 5.76·29-s + 3.09i·31-s + ⋯
L(s)  = 1  + (−0.394 − 0.394i)3-s + (−0.912 − 0.410i)5-s + (0.267 + 0.267i)7-s − 0.688i·9-s − 1.63·11-s + (0.875 − 0.875i)13-s + (0.197 + 0.521i)15-s + (−0.598 + 0.598i)17-s + 0.890i·19-s − 0.210i·21-s + (−0.137 + 0.137i)23-s + (0.663 + 0.748i)25-s + (−0.666 + 0.666i)27-s + 1.07·29-s + 0.556i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.287 - 0.957i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2784379959\)
\(L(\frac12)\) \(\approx\) \(0.2784379959\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.917i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.683 + 0.683i)T + 3iT^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 + (-3.15 + 3.15i)T - 13iT^{2} \)
17 \( 1 + (2.46 - 2.46i)T - 17iT^{2} \)
19 \( 1 - 3.87iT - 19T^{2} \)
23 \( 1 + (0.659 - 0.659i)T - 23iT^{2} \)
29 \( 1 - 5.76T + 29T^{2} \)
31 \( 1 - 3.09iT - 31T^{2} \)
37 \( 1 + (0.680 + 0.680i)T + 37iT^{2} \)
41 \( 1 + 7.39T + 41T^{2} \)
43 \( 1 + (-0.0846 - 0.0846i)T + 43iT^{2} \)
47 \( 1 + (2.32 + 2.32i)T + 47iT^{2} \)
53 \( 1 + (8.08 - 8.08i)T - 53iT^{2} \)
59 \( 1 - 7.94iT - 59T^{2} \)
61 \( 1 - 4.20iT - 61T^{2} \)
67 \( 1 + (7.82 - 7.82i)T - 67iT^{2} \)
71 \( 1 + 2.85iT - 71T^{2} \)
73 \( 1 + (-1.45 - 1.45i)T + 73iT^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + (-3.88 - 3.88i)T + 83iT^{2} \)
89 \( 1 - 1.96iT - 89T^{2} \)
97 \( 1 + (13.2 - 13.2i)T - 97iT^{2} \)
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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

−10.39108502969367939002100458277, −9.036686542283861433164757683455, −8.208366840255302130486202172619, −7.85496232658736966789924128690, −6.72130515919892962614018256887, −5.78973473159766945991490446810, −5.05348665139597024791313524188, −3.90169231640056658881805286783, −2.93585556630616301917328716184, −1.27714385325843218614238004229, 0.13700179547638672199207986740, 2.23872197590831536172781223242, 3.34854216089642096278485156701, 4.65124208663356510318683669732, 4.89458945035504811443721247906, 6.28963146435412417530280782506, 7.15134737494158957299285178471, 7.989431241775982103418840993717, 8.548594540170449069341583169857, 9.790035808884701637314823382879

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