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Properties

Label 2-1120-35.9-c1-0-38
Degree $2$
Conductor $1120$
Sign $-0.607 + 0.794i$
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  + (−1.21 + 0.699i)3-s + (1.87 + 1.21i)5-s + (0.374 − 2.61i)7-s + (−0.520 + 0.900i)9-s + (−1.87 − 3.25i)11-s − 4.96i·13-s + (−3.12 − 0.159i)15-s + (−5.30 + 3.06i)17-s + (−1.28 + 2.22i)19-s + (1.37 + 3.43i)21-s + (−7.81 − 4.51i)23-s + (2.04 + 4.56i)25-s − 5.65i·27-s − 3.74·29-s + (3.33 + 5.77i)31-s + ⋯
L(s)  = 1  + (−0.699 + 0.404i)3-s + (0.839 + 0.543i)5-s + (0.141 − 0.989i)7-s + (−0.173 + 0.300i)9-s + (−0.565 − 0.980i)11-s − 1.37i·13-s + (−0.807 − 0.0411i)15-s + (−1.28 + 0.743i)17-s + (−0.294 + 0.509i)19-s + (0.300 + 0.750i)21-s + (−1.62 − 0.940i)23-s + (0.409 + 0.912i)25-s − 1.08i·27-s − 0.694·29-s + (0.599 + 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.607 + 0.794i$
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: \((2,\ 1120,\ (\ :1/2),\ -0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4628820433\)
\(L(\frac12)\) \(\approx\) \(0.4628820433\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 - 1.21i)T \)
7 \( 1 + (-0.374 + 2.61i)T \)
good3 \( 1 + (1.21 - 0.699i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.87 + 3.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.96iT - 13T^{2} \)
17 \( 1 + (5.30 - 3.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.28 - 2.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.81 + 4.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 + (-3.33 - 5.77i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.798 - 0.460i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.36T + 41T^{2} \)
43 \( 1 + 6.42iT - 43T^{2} \)
47 \( 1 + (4.22 + 2.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.78 + 3.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.451 + 0.781i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.15 + 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.7 - 6.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.419T + 71T^{2} \)
73 \( 1 + (-2.32 + 1.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.17 - 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (-6.05 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.40iT - 97T^{2} \)
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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

−10.10326981032036199545571343213, −8.531269604133532379075696619975, −8.048271467381675074254424413273, −6.82830542063876493217761805485, −6.00493802439709428036350759567, −5.44465016666056655493223021958, −4.37522925925216804474672364674, −3.26705497304111380017184620601, −2.04344617277012470499864354861, −0.20266465803912496559911245935, 1.76659969755002509390230296966, 2.43818620151463957174657894516, 4.33783958650935548737286448783, 5.07497323449809899130168253210, 6.00819890894332410756835606311, 6.51268834455510071402226437619, 7.52469003458850846004298450664, 8.720937516843610904199451822433, 9.333971089857694002500191963870, 9.862997280713557092773813384641

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