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Properties

Label 2-1120-56.3-c1-0-14
Degree $2$
Conductor $1120$
Sign $0.612 - 0.790i$
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  + (1.84 + 1.06i)3-s + (0.5 + 0.866i)5-s + (2.17 − 1.50i)7-s + (0.759 + 1.31i)9-s + (−2.04 + 3.54i)11-s + 4.95·13-s + 2.12i·15-s + (2.09 + 1.20i)17-s + (−5.22 + 3.01i)19-s + (5.60 − 0.459i)21-s + (0.443 − 0.255i)23-s + (−0.499 + 0.866i)25-s − 3.14i·27-s − 3.08i·29-s + (−2.30 + 3.99i)31-s + ⋯
L(s)  = 1  + (1.06 + 0.613i)3-s + (0.223 + 0.387i)5-s + (0.822 − 0.569i)7-s + (0.253 + 0.438i)9-s + (−0.616 + 1.06i)11-s + 1.37·13-s + 0.548i·15-s + (0.507 + 0.292i)17-s + (−1.19 + 0.692i)19-s + (1.22 − 0.100i)21-s + (0.0923 − 0.0533i)23-s + (−0.0999 + 0.173i)25-s − 0.605i·27-s − 0.572i·29-s + (−0.414 + 0.717i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.645024741\)
\(L(\frac12)\) \(\approx\) \(2.645024741\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.17 + 1.50i)T \)
good3 \( 1 + (-1.84 - 1.06i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.04 - 3.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.95T + 13T^{2} \)
17 \( 1 + (-2.09 - 1.20i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.22 - 3.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.443 + 0.255i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.08iT - 29T^{2} \)
31 \( 1 + (2.30 - 3.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.64 + 4.99i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.81iT - 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + (0.698 + 1.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.79 + 3.92i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.82 + 5.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.33 + 2.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 - 6.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (-6.99 - 4.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.21 - 1.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.94iT - 83T^{2} \)
89 \( 1 + (3.81 - 2.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.67iT - 97T^{2} \)
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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

−9.909372787403558216848787366261, −9.186560051364972477840687151962, −8.113143909347499915932448080476, −7.916739354644438471966410980949, −6.67284485483793310428277777116, −5.67917392837629716571731890125, −4.37687469807944270372218816325, −3.89088393538001155787133688147, −2.70538826361204409233691028057, −1.63041063525442019544468453261, 1.18496604827483789155396303840, 2.31864650368108797334191154096, 3.17126016315372121325518316552, 4.43723471821410898591375862797, 5.58017864079306655866497719760, 6.25206088465765991501499746795, 7.63344976536831218290156529912, 8.116916115933679201360123121111, 8.837780606745516206205525649878, 9.203296912362102339161029779368

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