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Properties

Label 2-440-88.37-c1-0-40
Degree $2$
Conductor $440$
Sign $-0.857 - 0.514i$
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  + (−1.39 + 0.221i)2-s + (−1.53 − 0.5i)3-s + (1.90 − 0.618i)4-s + (−0.587 − 0.809i)5-s + (2.26 + 0.357i)6-s + (−1.61 − 4.97i)7-s + (−2.52 + 1.28i)8-s + (−0.309 − 0.224i)9-s + (1 + i)10-s + (3.30 + 0.309i)11-s − 3.23·12-s + (−3.07 + 4.23i)13-s + (3.36 + 6.59i)14-s + (0.5 + 1.53i)15-s + (3.23 − 2.35i)16-s + (2.11 − 1.53i)17-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)2-s + (−0.888 − 0.288i)3-s + (0.951 − 0.309i)4-s + (−0.262 − 0.361i)5-s + (0.922 + 0.146i)6-s + (−0.611 − 1.88i)7-s + (−0.891 + 0.453i)8-s + (−0.103 − 0.0748i)9-s + (0.316 + 0.316i)10-s + (0.995 + 0.0931i)11-s − 0.934·12-s + (−0.853 + 1.17i)13-s + (0.898 + 1.76i)14-s + (0.129 + 0.397i)15-s + (0.809 − 0.587i)16-s + (0.513 − 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.857 - 0.514i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.857 - 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0297121 + 0.107196i\)
\(L(\frac12)\) \(\approx\) \(0.0297121 + 0.107196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.221i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-3.30 - 0.309i)T \)
good3 \( 1 + (1.53 + 0.5i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.61 + 4.97i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.07 - 4.23i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.11 + 1.53i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.93 + 0.954i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.23T + 23T^{2} \)
29 \( 1 + (3.80 - 1.23i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (7.60 - 2.47i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.97 - 9.14i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.85iT - 43T^{2} \)
47 \( 1 + (2.14 - 6.60i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.25 + 5.85i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.587 + 0.190i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.62 + 3.61i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.32iT - 67T^{2} \)
71 \( 1 + (-5.61 + 4.08i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.354 + 1.08i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.7 + 9.23i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.40 + 1.92i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + (-11.3 - 8.28i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.56886871927306064518316216082, −9.701515836318628108767885427710, −8.962805747945367647291234381278, −7.57710117281552181018828470301, −6.87252525014796209996321367571, −6.40217793244819894543818858323, −4.84364789160047276269073359267, −3.57205266279577208516177003266, −1.37168640610331204808001783920, −0.11054186655158414483637481738, 2.28095039274724928638463802958, 3.40463296535418885041997449166, 5.43485892767149111691237070932, 5.97764883360857621987699231592, 6.99594052124004865127443758307, 8.281755833754083545887167721503, 8.974312544130104307436382959473, 9.939161407037704824782102345628, 10.66862242670159874590448287764, 11.63460601435747372250970706187

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