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Properties

Label 2-440-11.4-c1-0-5
Degree $2$
Conductor $440$
Sign $0.917 - 0.396i$
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  + (0.260 − 0.801i)3-s + (0.809 − 0.587i)5-s + (1.46 + 4.52i)7-s + (1.85 + 1.34i)9-s + (0.231 + 3.30i)11-s + (−3.06 − 2.22i)13-s + (−0.260 − 0.801i)15-s + (−4.03 + 2.92i)17-s + (0.964 − 2.96i)19-s + 4.01·21-s + 9.39·23-s + (0.309 − 0.951i)25-s + (3.60 − 2.62i)27-s + (−0.805 − 2.47i)29-s + (2.98 + 2.16i)31-s + ⋯
L(s)  = 1  + (0.150 − 0.462i)3-s + (0.361 − 0.262i)5-s + (0.555 + 1.70i)7-s + (0.617 + 0.448i)9-s + (0.0699 + 0.997i)11-s + (−0.848 − 0.616i)13-s + (−0.0672 − 0.207i)15-s + (−0.977 + 0.710i)17-s + (0.221 − 0.680i)19-s + 0.875·21-s + 1.95·23-s + (0.0618 − 0.190i)25-s + (0.694 − 0.504i)27-s + (−0.149 − 0.460i)29-s + (0.535 + 0.389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.917 - 0.396i$
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: \((2,\ 440,\ (\ :1/2),\ 0.917 - 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61079 + 0.333153i\)
\(L(\frac12)\) \(\approx\) \(1.61079 + 0.333153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.231 - 3.30i)T \)
good3 \( 1 + (-0.260 + 0.801i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.46 - 4.52i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.06 + 2.22i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.03 - 2.92i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.964 + 2.96i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 9.39T + 23T^{2} \)
29 \( 1 + (0.805 + 2.47i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.98 - 2.16i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.06 + 3.27i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.88 + 8.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (0.513 - 1.58i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.56 - 1.14i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0321 - 0.0988i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.21 + 5.96i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.99T + 67T^{2} \)
71 \( 1 + (-5.03 + 3.66i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.25 + 6.94i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.53 + 6.19i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.60 - 3.34i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (11.0 + 8.05i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.28370545379209527719669889369, −10.22317725684370155795346554401, −9.205402941007274873828311338060, −8.572813739001788176607386969485, −7.50221339341847800883488368767, −6.60274674011599696579628020247, −5.25220305140558457874910320355, −4.74633720825020161014496870072, −2.61550841554823985253451451127, −1.84183858800700416443801605238, 1.18190955653075260468001455583, 3.11902677859101332690357483945, 4.23746134367672528702105753216, 5.02085606002464017516420736220, 6.74505432467294661766939500486, 7.11549226418797774091726415108, 8.389443055749256609410829368577, 9.481515961132192877256714374362, 10.14103625552366521954125452856, 10.99927872071705980327985929443

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