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Properties

Label 2-440-11.3-c1-0-9
Degree $2$
Conductor $440$
Sign $-0.631 + 0.775i$
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  + (−0.400 − 1.23i)3-s + (−0.809 − 0.587i)5-s + (−0.0703 + 0.216i)7-s + (1.07 − 0.777i)9-s + (−3.12 − 1.12i)11-s + (2.70 − 1.96i)13-s + (−0.400 + 1.23i)15-s + (−4.89 − 3.55i)17-s + (−0.686 − 2.11i)19-s + 0.294·21-s − 3.47·23-s + (0.309 + 0.951i)25-s + (−4.52 − 3.29i)27-s + (0.00951 − 0.0292i)29-s + (−1.79 + 1.30i)31-s + ⋯
L(s)  = 1  + (−0.231 − 0.711i)3-s + (−0.361 − 0.262i)5-s + (−0.0266 + 0.0818i)7-s + (0.356 − 0.259i)9-s + (−0.940 − 0.339i)11-s + (0.750 − 0.545i)13-s + (−0.103 + 0.317i)15-s + (−1.18 − 0.862i)17-s + (−0.157 − 0.484i)19-s + 0.0643·21-s − 0.725·23-s + (0.0618 + 0.190i)25-s + (−0.871 − 0.633i)27-s + (0.00176 − 0.00543i)29-s + (−0.322 + 0.233i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.631 + 0.775i$
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: \((2,\ 440,\ (\ :1/2),\ -0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396341 - 0.834315i\)
\(L(\frac12)\) \(\approx\) \(0.396341 - 0.834315i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 + 1.12i)T \)
good3 \( 1 + (0.400 + 1.23i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.0703 - 0.216i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.70 + 1.96i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.89 + 3.55i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.686 + 2.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (-0.00951 + 0.0292i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.79 - 1.30i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.70 + 8.31i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.16 - 9.73i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.74T + 43T^{2} \)
47 \( 1 + (1.64 + 5.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.25 + 5.99i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.57 + 7.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.81 - 1.31i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.47T + 67T^{2} \)
71 \( 1 + (-12.1 - 8.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.13 - 3.48i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.40 + 6.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.23 + 0.895i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + (-6.74 + 4.89i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.04113696318588884041403664932, −9.930165679890972401266900206430, −8.843095947913578368749753570363, −7.986384311863863999215495393840, −7.11069537900287770330154205094, −6.17724108580945036629255910629, −5.09517817081897892977815296449, −3.84338510026078131222752664176, −2.35677416508424438763233200506, −0.58926391418376298880704739665, 2.10054498979154988159429955367, 3.80115755122288180724538833628, 4.48619453360452981436495177886, 5.69847082692282286471188278157, 6.79339727234877774123432868423, 7.84392766816820685175583628761, 8.735493804593452153605273178342, 9.883754112825847948318229731351, 10.60670284174318481302522726341, 11.14165813849762113583619400233

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