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Properties

Label 2-440-11.4-c1-0-4
Degree $2$
Conductor $440$
Sign $0.953 - 0.300i$
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  + (0.220 − 0.678i)3-s + (−0.809 + 0.587i)5-s + (0.116 + 0.357i)7-s + (2.01 + 1.46i)9-s + (−0.107 + 3.31i)11-s + (2.28 + 1.66i)13-s + (0.220 + 0.678i)15-s + (3.91 − 2.84i)17-s + (0.905 − 2.78i)19-s + 0.268·21-s + 3.77·23-s + (0.309 − 0.951i)25-s + (3.16 − 2.30i)27-s + (2.60 + 8.03i)29-s + (−6.50 − 4.72i)31-s + ⋯
L(s)  = 1  + (0.127 − 0.391i)3-s + (−0.361 + 0.262i)5-s + (0.0439 + 0.135i)7-s + (0.671 + 0.488i)9-s + (−0.0322 + 0.999i)11-s + (0.634 + 0.461i)13-s + (0.0569 + 0.175i)15-s + (0.949 − 0.689i)17-s + (0.207 − 0.639i)19-s + 0.0585·21-s + 0.786·23-s + (0.0618 − 0.190i)25-s + (0.609 − 0.443i)27-s + (0.484 + 1.49i)29-s + (−1.16 − 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.953 - 0.300i$
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: \((2,\ 440,\ (\ :1/2),\ 0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46861 + 0.226196i\)
\(L(\frac12)\) \(\approx\) \(1.46861 + 0.226196i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.107 - 3.31i)T \)
good3 \( 1 + (-0.220 + 0.678i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.116 - 0.357i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.28 - 1.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.91 + 2.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.905 + 2.78i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.77T + 23T^{2} \)
29 \( 1 + (-2.60 - 8.03i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.50 + 4.72i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.877 - 2.70i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.14 - 3.53i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + (-0.800 + 2.46i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.0394 - 0.0286i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.509 + 1.56i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.03 - 5.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + (11.4 - 8.30i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.158 - 0.488i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.5 + 7.63i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.21 - 1.60i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (13.0 + 9.44i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.22201409007987224620879359672, −10.27971267349164670316100503701, −9.393887982678359082115641878374, −8.353740188481386864660898370344, −7.24003586999556749976702683465, −6.93152075927988653272943307354, −5.34585248989687557865770557740, −4.36487525814323683909402969779, −2.98980563497984996142581005561, −1.54111789333844679433075121655, 1.13939369424324292749274827464, 3.30235383509271494207009931264, 3.98863659188305576963984000261, 5.34337861913319977472289856398, 6.29612804906253895589668566744, 7.56188208330529609097374532968, 8.386615599368587577261994896533, 9.247481451479194087303523341253, 10.27607151729374315926599900131, 10.92047977419475519988018450614

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