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Properties

Label 2-1120-140.139-c1-0-31
Degree $2$
Conductor $1120$
Sign $0.932 + 0.360i$
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  − 0.568i·3-s + (2.03 + 0.918i)5-s + (1.76 − 1.96i)7-s + 2.67·9-s + 0.417i·11-s + 2.13·13-s + (0.521 − 1.15i)15-s + 0.314·17-s − 2.19·19-s + (−1.11 − 1.00i)21-s + 0.992·23-s + (3.31 + 3.74i)25-s − 3.22i·27-s − 4.35·29-s − 4.22·31-s + ⋯
L(s)  = 1  − 0.328i·3-s + (0.911 + 0.410i)5-s + (0.667 − 0.744i)7-s + 0.892·9-s + 0.125i·11-s + 0.592·13-s + (0.134 − 0.299i)15-s + 0.0761·17-s − 0.503·19-s + (−0.244 − 0.219i)21-s + 0.206·23-s + (0.662 + 0.748i)25-s − 0.620i·27-s − 0.808·29-s − 0.759·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.252685735\)
\(L(\frac12)\) \(\approx\) \(2.252685735\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 - 0.918i)T \)
7 \( 1 + (-1.76 + 1.96i)T \)
good3 \( 1 + 0.568iT - 3T^{2} \)
11 \( 1 - 0.417iT - 11T^{2} \)
13 \( 1 - 2.13T + 13T^{2} \)
17 \( 1 - 0.314T + 17T^{2} \)
19 \( 1 + 2.19T + 19T^{2} \)
23 \( 1 - 0.992T + 23T^{2} \)
29 \( 1 + 4.35T + 29T^{2} \)
31 \( 1 + 4.22T + 31T^{2} \)
37 \( 1 - 6.83iT - 37T^{2} \)
41 \( 1 + 3.82iT - 41T^{2} \)
43 \( 1 - 6.04T + 43T^{2} \)
47 \( 1 + 4.98iT - 47T^{2} \)
53 \( 1 - 7.14iT - 53T^{2} \)
59 \( 1 + 9.50T + 59T^{2} \)
61 \( 1 + 0.783iT - 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 6.55iT - 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + 5.49iT - 83T^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 - 0.314T + 97T^{2} \)
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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

−9.856076060450666099617489863803, −9.059702132997446900635043690865, −7.997007796694490655038391983141, −7.20936712268591878753726850055, −6.56965303637134843003126104873, −5.59670307857907670225243432917, −4.57058922422847916306899253667, −3.59806732328877804353168793081, −2.13081032133467644637539895407, −1.25990240272732819513237289941, 1.39373678713094472316336090255, 2.33338650091995640259224522298, 3.81566640011200618666784937535, 4.80232790758933140050227238747, 5.56384192538008171602661915312, 6.33724582861870103808113090292, 7.45632058401285135661574571688, 8.424140080735181306767074265339, 9.180549583563878921463171244820, 9.688335742449873021444181891961

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