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Properties

Label 2-1120-40.29-c1-0-31
Degree $2$
Conductor $1120$
Sign $0.970 + 0.240i$
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  + 2.96·3-s + (2.18 − 0.458i)5-s − i·7-s + 5.80·9-s + 0.338i·11-s − 2.66·13-s + (6.49 − 1.36i)15-s + 3.60i·17-s − 7.58i·19-s − 2.96i·21-s + 1.51i·23-s + (4.57 − 2.00i)25-s + 8.30·27-s + 9.39i·29-s − 3.04·31-s + ⋯
L(s)  = 1  + 1.71·3-s + (0.978 − 0.205i)5-s − 0.377i·7-s + 1.93·9-s + 0.102i·11-s − 0.738·13-s + (1.67 − 0.351i)15-s + 0.874i·17-s − 1.74i·19-s − 0.647i·21-s + 0.316i·23-s + (0.915 − 0.401i)25-s + 1.59·27-s + 1.74i·29-s − 0.546·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.970 + 0.240i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.970 + 0.240i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.380962145\)
\(L(\frac12)\) \(\approx\) \(3.380962145\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 + 0.458i)T \)
7 \( 1 + iT \)
good3 \( 1 - 2.96T + 3T^{2} \)
11 \( 1 - 0.338iT - 11T^{2} \)
13 \( 1 + 2.66T + 13T^{2} \)
17 \( 1 - 3.60iT - 17T^{2} \)
19 \( 1 + 7.58iT - 19T^{2} \)
23 \( 1 - 1.51iT - 23T^{2} \)
29 \( 1 - 9.39iT - 29T^{2} \)
31 \( 1 + 3.04T + 31T^{2} \)
37 \( 1 - 5.70T + 37T^{2} \)
41 \( 1 + 6.41T + 41T^{2} \)
43 \( 1 + 7.73T + 43T^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 - 2.35T + 53T^{2} \)
59 \( 1 - 8.53iT - 59T^{2} \)
61 \( 1 - 2.39iT - 61T^{2} \)
67 \( 1 - 6.93T + 67T^{2} \)
71 \( 1 + 0.174T + 71T^{2} \)
73 \( 1 - 9.77iT - 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + 7.09iT - 97T^{2} \)
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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

−9.762203641069900940584589826565, −8.792444763046152821763114176937, −8.545076423071371486489975058538, −7.24423068674142405380448646343, −6.86242232525111437239153636161, −5.38148220649271975903676848111, −4.46276294713655036376292487060, −3.32774332727347983498562595254, −2.46647229582626300165406421942, −1.52619556685562821274509152971, 1.74924291798216788381907539882, 2.51691509996731509658658072013, 3.33360142824309162612550632490, 4.49688022379654089820597250833, 5.65109311236161780360484856379, 6.63018804403982889766709212014, 7.66149651511418471042124526931, 8.223555569838755478199197688923, 9.139294285530308000411206454389, 9.810970419183823899849705046839

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