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Properties

Label 2-440-11.5-c1-0-7
Degree $2$
Conductor $440$
Sign $-0.674 + 0.738i$
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  + (−2.59 − 1.88i)3-s + (0.309 − 0.951i)5-s + (3.97 − 2.88i)7-s + (2.26 + 6.95i)9-s + (2.80 − 1.76i)11-s + (−0.248 − 0.765i)13-s + (−2.59 + 1.88i)15-s + (0.570 − 1.75i)17-s + (−4.54 − 3.30i)19-s − 15.7·21-s − 1.60·23-s + (−0.809 − 0.587i)25-s + (4.28 − 13.1i)27-s + (−6.36 + 4.62i)29-s + (1.19 + 3.68i)31-s + ⋯
L(s)  = 1  + (−1.50 − 1.09i)3-s + (0.138 − 0.425i)5-s + (1.50 − 1.09i)7-s + (0.753 + 2.31i)9-s + (0.846 − 0.532i)11-s + (−0.0689 − 0.212i)13-s + (−0.670 + 0.487i)15-s + (0.138 − 0.426i)17-s + (−1.04 − 0.757i)19-s − 3.44·21-s − 0.334·23-s + (−0.161 − 0.117i)25-s + (0.824 − 2.53i)27-s + (−1.18 + 0.858i)29-s + (0.214 + 0.661i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.674 + 0.738i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.674 + 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.385392 - 0.874304i\)
\(L(\frac12)\) \(\approx\) \(0.385392 - 0.874304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-2.80 + 1.76i)T \)
good3 \( 1 + (2.59 + 1.88i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-3.97 + 2.88i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.248 + 0.765i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.570 + 1.75i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.54 + 3.30i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 + (6.36 - 4.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.19 - 3.68i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.40 + 3.92i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.23 - 3.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.05T + 43T^{2} \)
47 \( 1 + (6.99 + 5.08i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.41 - 7.44i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.14 - 2.28i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.04 + 6.29i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5.52T + 67T^{2} \)
71 \( 1 + (3.38 - 10.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.26 - 1.64i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.902 + 2.77i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.598 + 1.84i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 7.32T + 89T^{2} \)
97 \( 1 + (-2.52 - 7.75i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.21151767773396486391063292449, −10.33497825865254040979542699406, −8.768764593814427294407826804940, −7.73311209385820335850460642019, −7.07914420837595787252308674959, −6.08345343948372667415589680304, −5.09570957481957446874483806931, −4.31322412774174602908963949522, −1.75781972340813314395795195879, −0.78808669248002906598628828555, 1.86624588444213050249406635990, 4.02561054863373431753535070565, 4.71606059191919758766852906431, 5.80886072157601691080282882071, 6.28900423980946767499543735024, 7.80590549214746545989019301197, 9.015750117547951588693747700981, 9.837548855272472013904358628975, 10.71663227726544979426818820335, 11.53853224265602996857300967633

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