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Properties

Label 2-2240-140.139-c1-0-75
Degree $2$
Conductor $2240$
Sign $-0.500 + 0.865i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  − 3.06i·3-s + (1.39 + 1.75i)5-s + (2.46 − 0.972i)7-s − 6.38·9-s − 5.46i·11-s + 1.22·13-s + (5.36 − 4.25i)15-s + 0.813·17-s + 6.68·19-s + (−2.97 − 7.53i)21-s + 3.51·23-s + (−1.13 + 4.86i)25-s + 10.3i·27-s + 8.09·29-s − 9.00·31-s + ⋯
L(s)  = 1  − 1.76i·3-s + (0.621 + 0.783i)5-s + (0.930 − 0.367i)7-s − 2.12·9-s − 1.64i·11-s + 0.340·13-s + (1.38 − 1.09i)15-s + 0.197·17-s + 1.53·19-s + (−0.650 − 1.64i)21-s + 0.733·23-s + (−0.227 + 0.973i)25-s + 1.99i·27-s + 1.50·29-s − 1.61·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.500 + 0.865i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.281283565\)
\(L(\frac12)\) \(\approx\) \(2.281283565\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 - 1.75i)T \)
7 \( 1 + (-2.46 + 0.972i)T \)
good3 \( 1 + 3.06iT - 3T^{2} \)
11 \( 1 + 5.46iT - 11T^{2} \)
13 \( 1 - 1.22T + 13T^{2} \)
17 \( 1 - 0.813T + 17T^{2} \)
19 \( 1 - 6.68T + 19T^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 - 8.09T + 29T^{2} \)
31 \( 1 + 9.00T + 31T^{2} \)
37 \( 1 + 4.61iT - 37T^{2} \)
41 \( 1 - 2.59iT - 41T^{2} \)
43 \( 1 + 1.61T + 43T^{2} \)
47 \( 1 - 5.97iT - 47T^{2} \)
53 \( 1 - 1.51iT - 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 3.16iT - 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 + 2.19iT - 79T^{2} \)
83 \( 1 + 7.88iT - 83T^{2} \)
89 \( 1 + 1.19iT - 89T^{2} \)
97 \( 1 - 0.813T + 97T^{2} \)
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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

−8.567761854599377124692696442605, −7.76051263438649289584185683564, −7.33778604109722567582112860902, −6.47918791131879071341674612163, −5.85280062630077226039390223503, −5.18576560244862522156979187218, −3.41441232814640650702845216674, −2.77192791554020113486581378483, −1.58978171203738764971052546022, −0.880029936094559735948487448997, 1.44891186598341771562678101034, 2.64854324365496582258331219194, 3.81997823737415794618300589071, 4.72194178004354617537972210581, 5.08473152178137285339653708860, 5.63892236618401057475408703449, 6.97947725966479295063118194990, 8.063708773250414042367825259749, 8.783472369435209108327132179625, 9.432702746150607712303616891991

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