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Properties

Label 2-440-11.3-c1-0-11
Degree $2$
Conductor $440$
Sign $-0.961 + 0.274i$
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  + (−1.01 − 3.13i)3-s + (0.809 + 0.587i)5-s + (1.08 − 3.34i)7-s + (−6.34 + 4.61i)9-s + (−1.91 − 2.70i)11-s + (3.45 − 2.51i)13-s + (1.01 − 3.13i)15-s + (−1.28 − 0.934i)17-s + (1.71 + 5.27i)19-s − 11.5·21-s − 6.39·23-s + (0.309 + 0.951i)25-s + (12.9 + 9.37i)27-s + (−0.117 + 0.362i)29-s + (0.615 − 0.446i)31-s + ⋯
L(s)  = 1  + (−0.587 − 1.80i)3-s + (0.361 + 0.262i)5-s + (0.410 − 1.26i)7-s + (−2.11 + 1.53i)9-s + (−0.577 − 0.816i)11-s + (0.958 − 0.696i)13-s + (0.262 − 0.808i)15-s + (−0.311 − 0.226i)17-s + (0.393 + 1.21i)19-s − 2.52·21-s − 1.33·23-s + (0.0618 + 0.190i)25-s + (2.48 + 1.80i)27-s + (−0.0218 + 0.0673i)29-s + (0.110 − 0.0802i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.961 + 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143203 - 1.02513i\)
\(L(\frac12)\) \(\approx\) \(0.143203 - 1.02513i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 + 2.70i)T \)
good3 \( 1 + (1.01 + 3.13i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-1.08 + 3.34i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.45 + 2.51i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.28 + 0.934i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.71 - 5.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.39T + 23T^{2} \)
29 \( 1 + (0.117 - 0.362i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.615 + 0.446i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.448 - 1.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.89 + 5.82i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.19T + 43T^{2} \)
47 \( 1 + (-1.33 - 4.11i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.62 + 4.08i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.92 + 12.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.19 - 5.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + (5.95 + 4.32i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.17 + 6.68i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.65 + 4.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.69 - 6.31i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.451T + 89T^{2} \)
97 \( 1 + (-2.24 + 1.63i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.87768721254652628370802816689, −10.20387441894608882420941149224, −8.289940565659168442409560182379, −7.942733595131854668705025693281, −6.97113047829139690429363037005, −6.11319712675614703092530729658, −5.37608207519604590110089995607, −3.50251028289401625836189004785, −1.91685447993736165627199061248, −0.70496052822937729456297940658, 2.40481881395112553175317911981, 3.93086092103593600077359684061, 4.90296778112341571371327309282, 5.52086166912680375474873423836, 6.48088556879947648679156387136, 8.389884177419024308808581863154, 9.022282347988513799299863478221, 9.760180404394830778003844243836, 10.53512227619688109599860807539, 11.54236318678935268493523940224

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