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Properties

Label 2-440-11.4-c1-0-8
Degree $2$
Conductor $440$
Sign $0.276 + 0.960i$
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  + (0.0911 − 0.280i)3-s + (−0.809 + 0.587i)5-s + (−1.35 − 4.17i)7-s + (2.35 + 1.71i)9-s + (3.31 + 0.189i)11-s + (−4.82 − 3.50i)13-s + (0.0911 + 0.280i)15-s + (1.34 − 0.980i)17-s + (2.37 − 7.31i)19-s − 1.29·21-s − 0.904·23-s + (0.309 − 0.951i)25-s + (1.41 − 1.02i)27-s + (−1.46 − 4.50i)29-s + (4.14 + 3.01i)31-s + ⋯
L(s)  = 1  + (0.0526 − 0.161i)3-s + (−0.361 + 0.262i)5-s + (−0.512 − 1.57i)7-s + (0.785 + 0.570i)9-s + (0.998 + 0.0572i)11-s + (−1.33 − 0.972i)13-s + (0.0235 + 0.0724i)15-s + (0.327 − 0.237i)17-s + (0.545 − 1.67i)19-s − 0.282·21-s − 0.188·23-s + (0.0618 − 0.190i)25-s + (0.271 − 0.197i)27-s + (−0.271 − 0.836i)29-s + (0.744 + 0.541i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.276 + 0.960i$
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: \((2,\ 440,\ (\ :1/2),\ 0.276 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.989371 - 0.744464i\)
\(L(\frac12)\) \(\approx\) \(0.989371 - 0.744464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 0.189i)T \)
good3 \( 1 + (-0.0911 + 0.280i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (1.35 + 4.17i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.82 + 3.50i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.34 + 0.980i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.37 + 7.31i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.904T + 23T^{2} \)
29 \( 1 + (1.46 + 4.50i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.14 - 3.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.0571 + 0.175i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.810 - 2.49i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.59T + 43T^{2} \)
47 \( 1 + (0.239 - 0.738i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.76 - 5.64i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.47 + 10.7i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (10.6 - 7.75i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.79T + 67T^{2} \)
71 \( 1 + (-5.63 + 4.09i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.94 - 12.1i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.11 - 5.89i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.31 - 2.40i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.466T + 89T^{2} \)
97 \( 1 + (5.74 + 4.17i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.79131685459330489717206647785, −10.09182483246723354686896640189, −9.394466574214001226346723877867, −7.79731463073058175632531424113, −7.29839897815584585365155040716, −6.61494592633411813656455470316, −4.91568378151452990099259827733, −4.07232163517642253544778752269, −2.82356285330956316052124726728, −0.827391488150636135195030187681, 1.81522504338859554757022467939, 3.36918626594579518885347181966, 4.43136029882681674651338465277, 5.67603466221223654703523170840, 6.57936174173825130002478185114, 7.63719031331433006923170695713, 8.876327801369451934842630194657, 9.418381344918478971483958862301, 10.11724898152441758839590385911, 11.73583799996150363083052608492

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