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Properties

Label 2-440-11.4-c1-0-1
Degree $2$
Conductor $440$
Sign $-0.676 - 0.736i$
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  + (−0.507 + 1.56i)3-s + (−0.809 + 0.587i)5-s + (1.04 + 3.21i)7-s + (0.248 + 0.180i)9-s + (3.04 − 1.30i)11-s + (−3.81 − 2.76i)13-s + (−0.507 − 1.56i)15-s + (−6.11 + 4.44i)17-s + (−0.689 + 2.12i)19-s − 5.54·21-s − 0.0373·23-s + (0.309 − 0.951i)25-s + (−4.39 + 3.18i)27-s + (1.96 + 6.05i)29-s + (0.282 + 0.205i)31-s + ⋯
L(s)  = 1  + (−0.292 + 0.901i)3-s + (−0.361 + 0.262i)5-s + (0.394 + 1.21i)7-s + (0.0827 + 0.0600i)9-s + (0.918 − 0.394i)11-s + (−1.05 − 0.767i)13-s + (−0.130 − 0.402i)15-s + (−1.48 + 1.07i)17-s + (−0.158 + 0.486i)19-s − 1.20·21-s − 0.00778·23-s + (0.0618 − 0.190i)25-s + (−0.844 + 0.613i)27-s + (0.365 + 1.12i)29-s + (0.0506 + 0.0368i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.676 - 0.736i$
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: \((2,\ 440,\ (\ :1/2),\ -0.676 - 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424461 + 0.966778i\)
\(L(\frac12)\) \(\approx\) \(0.424461 + 0.966778i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 + 1.30i)T \)
good3 \( 1 + (0.507 - 1.56i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.04 - 3.21i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.81 + 2.76i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.11 - 4.44i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.689 - 2.12i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.0373T + 23T^{2} \)
29 \( 1 + (-1.96 - 6.05i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.282 - 0.205i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.04 + 6.30i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.609 - 1.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + (1.96 - 6.05i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.33 + 5.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.959 + 2.95i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.45 + 6.86i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 + (0.193 - 0.140i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.12 - 12.7i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.73 - 4.16i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.72 + 7.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.87T + 89T^{2} \)
97 \( 1 + (-12.7 - 9.27i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.18180878124079555176074997706, −10.76173542122257772514806721242, −9.642831076911514970664149062339, −8.859272812049551153441862577445, −7.978465556811940355415117592976, −6.67807522269725356952172676894, −5.61643628501125351798319807970, −4.70230370888765094474162315702, −3.67851379222933178352743322891, −2.18921705627951998226304091303, 0.69658551534924840307116105096, 2.13839265976743314472399681472, 4.15977179802465807367194173818, 4.69834595309504213696910553789, 6.53239760282823330810291448018, 7.04152745010776031383858266509, 7.69744753760038629258462142230, 9.027988622803722720440777891658, 9.820699906434691999821540866366, 11.07305137119058094131678857383

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