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Properties

Label 2-1120-35.9-c1-0-25
Degree $2$
Conductor $1120$
Sign $-0.160 + 0.987i$
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  + (−0.733 + 0.423i)3-s + (−2.17 + 0.498i)5-s + (−1.14 + 2.38i)7-s + (−1.14 + 1.97i)9-s + (−0.573 − 0.993i)11-s + 1.58i·13-s + (1.38 − 1.28i)15-s + (−3.61 + 2.08i)17-s + (3.82 − 6.62i)19-s + (−0.166 − 2.23i)21-s + (−4.27 − 2.46i)23-s + (4.50 − 2.17i)25-s − 4.47i·27-s + 5.62·29-s + (2.31 + 4.01i)31-s + ⋯
L(s)  = 1  + (−0.423 + 0.244i)3-s + (−0.974 + 0.222i)5-s + (−0.434 + 0.900i)7-s + (−0.380 + 0.658i)9-s + (−0.172 − 0.299i)11-s + 0.438i·13-s + (0.358 − 0.332i)15-s + (−0.876 + 0.506i)17-s + (0.876 − 1.51i)19-s + (−0.0362 − 0.487i)21-s + (−0.890 − 0.514i)23-s + (0.900 − 0.434i)25-s − 0.861i·27-s + 1.04·29-s + (0.415 + 0.720i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.160 + 0.987i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2213531419\)
\(L(\frac12)\) \(\approx\) \(0.2213531419\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 - 0.498i)T \)
7 \( 1 + (1.14 - 2.38i)T \)
good3 \( 1 + (0.733 - 0.423i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.573 + 0.993i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.58iT - 13T^{2} \)
17 \( 1 + (3.61 - 2.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.82 + 6.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.27 + 2.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.62T + 29T^{2} \)
31 \( 1 + (-2.31 - 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.03 + 0.595i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.457T + 41T^{2} \)
43 \( 1 + 4.02iT - 43T^{2} \)
47 \( 1 + (10.2 + 5.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.27 + 0.738i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.38 + 12.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.66 - 9.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.89 + 4.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.40T + 71T^{2} \)
73 \( 1 + (2.18 - 1.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.29 + 5.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.85iT - 83T^{2} \)
89 \( 1 + (4.88 - 8.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−9.609527574013283023039464695881, −8.590140570347360906209334305860, −8.211270110730762839013549035209, −6.95153081843797548371641660186, −6.32207648553693111911908921842, −5.17624770585157812367608973738, −4.52596332662834478667668757608, −3.26074364981523098519663991165, −2.34780726774839877063645336752, −0.11780035474349320592287301848, 1.12241250135826509574331405902, 3.06832266726286134125520867965, 3.88184830695129393185557562961, 4.79930990350410679785619805296, 5.97127344876182805520643238443, 6.73275328533349808396985872789, 7.65336592826922027070430768897, 8.149093207207890148859687284570, 9.357128449749461467320253218106, 10.04321582627908913933279549450

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