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Properties

Label 2-440-11.4-c1-0-10
Degree $2$
Conductor $440$
Sign $-0.338 + 0.940i$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
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  + (0.584 − 1.79i)3-s + (0.809 − 0.587i)5-s + (−0.846 − 2.60i)7-s + (−0.464 − 0.337i)9-s + (−2.57 − 2.09i)11-s + (−0.159 − 0.115i)13-s + (−0.584 − 1.79i)15-s + (1.18 − 0.862i)17-s + (−0.828 + 2.55i)19-s − 5.18·21-s + 1.81·23-s + (0.309 − 0.951i)25-s + (3.70 − 2.69i)27-s + (0.426 + 1.31i)29-s + (−4.77 − 3.47i)31-s + ⋯
L(s)  = 1  + (0.337 − 1.03i)3-s + (0.361 − 0.262i)5-s + (−0.320 − 0.985i)7-s + (−0.154 − 0.112i)9-s + (−0.775 − 0.631i)11-s + (−0.0441 − 0.0320i)13-s + (−0.150 − 0.464i)15-s + (0.288 − 0.209i)17-s + (−0.190 + 0.585i)19-s − 1.13·21-s + 0.378·23-s + (0.0618 − 0.190i)25-s + (0.713 − 0.518i)27-s + (0.0791 + 0.243i)29-s + (−0.857 − 0.623i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.338 + 0.940i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.338 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855748 - 1.21775i\)
\(L(\frac12)\) \(\approx\) \(0.855748 - 1.21775i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (2.57 + 2.09i)T \)
good3 \( 1 + (-0.584 + 1.79i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.846 + 2.60i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.159 + 0.115i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.18 + 0.862i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.828 - 2.55i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.81T + 23T^{2} \)
29 \( 1 + (-0.426 - 1.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.77 + 3.47i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.41 - 4.35i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.381 + 1.17i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 + (-2.87 + 8.84i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.17 + 1.58i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.39 - 13.5i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.10 - 4.43i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + (-5.56 + 4.04i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.22 - 9.92i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.51 + 1.10i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.68 + 3.40i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + (-14.8 - 10.7i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.70387505763731089837449278916, −10.08566747149948935862979898679, −8.886532140380702684497504505411, −7.908430528965045707694990263437, −7.28460011291192184786250525487, −6.31280851680218110941414586587, −5.19111308583470348567584130198, −3.74815683717354459624118449064, −2.39049609642751341267453345414, −0.939776959329316162321523106106, 2.31331158316361531019450438034, 3.35366594447460076749370825928, 4.66298778938805579195199852774, 5.52595360751787004337388651398, 6.67269446677948377193377993713, 7.86349171762259707101817567396, 9.095416453972448228999547724351, 9.467366664406831245494236099285, 10.40952771748220478610783246206, 11.10464389519060028682236788786

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