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Properties

Label 2-1120-56.19-c1-0-12
Degree $2$
Conductor $1120$
Sign $0.965 - 0.259i$
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  + (−0.725 + 0.418i)3-s + (0.5 − 0.866i)5-s + (−2.36 + 1.17i)7-s + (−1.14 + 1.99i)9-s + (−2.98 − 5.17i)11-s + 2.87·13-s + 0.837i·15-s + (2.07 − 1.19i)17-s + (4.05 + 2.34i)19-s + (1.22 − 1.84i)21-s + (5.81 + 3.35i)23-s + (−0.499 − 0.866i)25-s − 4.43i·27-s − 1.31i·29-s + (4.34 + 7.53i)31-s + ⋯
L(s)  = 1  + (−0.418 + 0.241i)3-s + (0.223 − 0.387i)5-s + (−0.895 + 0.445i)7-s + (−0.382 + 0.663i)9-s + (−0.901 − 1.56i)11-s + 0.797·13-s + 0.216i·15-s + (0.503 − 0.290i)17-s + (0.930 + 0.537i)19-s + (0.267 − 0.403i)21-s + (1.21 + 0.700i)23-s + (−0.0999 − 0.173i)25-s − 0.854i·27-s − 0.243i·29-s + (0.780 + 1.35i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.965 - 0.259i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.233665110\)
\(L(\frac12)\) \(\approx\) \(1.233665110\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 - 1.17i)T \)
good3 \( 1 + (0.725 - 0.418i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.98 + 5.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 + (-2.07 + 1.19i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.05 - 2.34i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.81 - 3.35i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.31iT - 29T^{2} \)
31 \( 1 + (-4.34 - 7.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.26 - 1.88i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.79iT - 41T^{2} \)
43 \( 1 - 6.73T + 43T^{2} \)
47 \( 1 + (-0.874 + 1.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0994 + 0.0574i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.61 - 1.50i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.40 - 7.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.83 - 4.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + (-8.69 + 5.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.9 - 6.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−9.944982703607321940989193838162, −8.974751369280079154560586572650, −8.385715310782566645485929070038, −7.46430520329799981468330749432, −6.13655903114962481699917669614, −5.65295827720340524774229944624, −4.99961732776433944364377636640, −3.43728261506062665599207912437, −2.78570563325156173794011327053, −0.906641139771559530203285894084, 0.825190278538491529377142550354, 2.53321849092719650768697852460, 3.45410283336622545095222595968, 4.64422682711158611967568998655, 5.70737118121626960929834006273, 6.50836247389356731289750091856, 7.14430297817289286985061251219, 7.956851538405604264681064835550, 9.326420754810632382832722448510, 9.683240677903028080833195822737

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