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Properties

Label 2-440-11.4-c1-0-3
Degree $2$
Conductor $440$
Sign $0.387 - 0.921i$
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  + (−0.535 + 1.64i)3-s + (0.809 − 0.587i)5-s + (−0.386 − 1.19i)7-s + (−0.00499 − 0.00363i)9-s + (2.45 + 2.22i)11-s + (2.02 + 1.47i)13-s + (0.535 + 1.64i)15-s + (2.91 − 2.11i)17-s + (−2.18 + 6.71i)19-s + 2.17·21-s − 2.20·23-s + (0.309 − 0.951i)25-s + (−4.19 + 3.05i)27-s + (0.834 + 2.56i)29-s + (2.29 + 1.66i)31-s + ⋯
L(s)  = 1  + (−0.309 + 0.952i)3-s + (0.361 − 0.262i)5-s + (−0.146 − 0.450i)7-s + (−0.00166 − 0.00121i)9-s + (0.740 + 0.671i)11-s + (0.562 + 0.408i)13-s + (0.138 + 0.425i)15-s + (0.707 − 0.513i)17-s + (−0.500 + 1.53i)19-s + 0.473·21-s − 0.460·23-s + (0.0618 − 0.190i)25-s + (−0.808 + 0.587i)27-s + (0.154 + 0.476i)29-s + (0.411 + 0.299i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.387 - 0.921i$
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: \((2,\ 440,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15929 + 0.769881i\)
\(L(\frac12)\) \(\approx\) \(1.15929 + 0.769881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 2.22i)T \)
good3 \( 1 + (0.535 - 1.64i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.386 + 1.19i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.02 - 1.47i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.91 + 2.11i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.18 - 6.71i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 + (-0.834 - 2.56i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.29 - 1.66i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.49 + 7.67i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.19 - 9.82i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.88T + 43T^{2} \)
47 \( 1 + (-0.493 + 1.51i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.43 + 1.04i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.35 + 4.15i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.74 + 4.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.43T + 67T^{2} \)
71 \( 1 + (-3.75 + 2.72i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.628 + 1.93i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (13.5 + 9.85i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.55 + 3.30i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.72T + 89T^{2} \)
97 \( 1 + (15.4 + 11.2i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.13620813459775029205445701720, −10.16243027173268638562908214230, −9.793477829060645920194561136737, −8.801383179370513332244003266301, −7.61914410884214982885083406237, −6.49844408431845431702320100385, −5.47321951716511496229185875931, −4.39290199715100833502566944944, −3.65659104034544428698115938858, −1.64210465673872578692151492699, 1.05589850295381788949596022080, 2.54227896657092694746783781270, 3.94365626076080088534808097841, 5.59952360064470531957493834887, 6.30271262476703427865712411373, 7.03711770208319005879040545415, 8.210896593560664441827202349707, 9.048551997565837718961313503074, 10.11327099751453913813023551859, 11.11844526680782858224900945424

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