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Properties

Label 2-1120-5.4-c1-0-9
Degree $2$
Conductor $1120$
Sign $-0.735 - 0.677i$
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  + 2.19i·3-s + (1.64 + 1.51i)5-s − i·7-s − 1.83·9-s − 1.37·11-s − 2.74i·13-s + (−3.32 + 3.61i)15-s + 6.94i·17-s − 1.29·19-s + 2.19·21-s + 8.31i·23-s + (0.412 + 4.98i)25-s + 2.56i·27-s + 8.40·29-s − 9.49·31-s + ⋯
L(s)  = 1  + 1.26i·3-s + (0.735 + 0.677i)5-s − 0.377i·7-s − 0.610·9-s − 0.414·11-s − 0.760i·13-s + (−0.859 + 0.933i)15-s + 1.68i·17-s − 0.296·19-s + 0.479·21-s + 1.73i·23-s + (0.0825 + 0.996i)25-s + 0.494i·27-s + 1.56·29-s − 1.70·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.611478081\)
\(L(\frac12)\) \(\approx\) \(1.611478081\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 1.51i)T \)
7 \( 1 + iT \)
good3 \( 1 - 2.19iT - 3T^{2} \)
11 \( 1 + 1.37T + 11T^{2} \)
13 \( 1 + 2.74iT - 13T^{2} \)
17 \( 1 - 6.94iT - 17T^{2} \)
19 \( 1 + 1.29T + 19T^{2} \)
23 \( 1 - 8.31iT - 23T^{2} \)
29 \( 1 - 8.40T + 29T^{2} \)
31 \( 1 + 9.49T + 31T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 + 5.30T + 41T^{2} \)
43 \( 1 + 7.83iT - 43T^{2} \)
47 \( 1 - 3.48iT - 47T^{2} \)
53 \( 1 + 6.13iT - 53T^{2} \)
59 \( 1 + 6.26T + 59T^{2} \)
61 \( 1 - 6.59T + 61T^{2} \)
67 \( 1 - 1.66iT - 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + 2.13iT - 73T^{2} \)
79 \( 1 - 5.45T + 79T^{2} \)
83 \( 1 + 2.54iT - 83T^{2} \)
89 \( 1 + 1.43T + 89T^{2} \)
97 \( 1 + 9.69iT - 97T^{2} \)
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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

−10.24181196204751906784474950279, −9.611897094794763487684317714646, −8.675250926318327928869972303642, −7.72962476325490237343441677090, −6.71333165339048538161330033256, −5.70823521496195519664235029503, −5.08869663698675835505780575385, −3.82698563331389767977512243193, −3.26989393520487702280468685592, −1.80458701356395642858510075695, 0.69952358444529258443580986823, 1.97705636880489886589171488896, 2.69497627617585194028274129723, 4.52715536072389343182417756830, 5.28158059617037574602734546512, 6.39007345461125351452102554681, 6.84882375199903376627807614150, 7.87542219422030990317117328777, 8.672405552679703914151064357681, 9.342063918001430599838127448037

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