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Properties

Label 2-440-11.5-c1-0-1
Degree $2$
Conductor $440$
Sign $0.964 - 0.265i$
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  + (−2.68 − 1.94i)3-s + (−0.309 + 0.951i)5-s + (−0.150 + 0.109i)7-s + (2.46 + 7.59i)9-s + (−3.31 + 0.00161i)11-s + (0.931 + 2.86i)13-s + (2.68 − 1.94i)15-s + (1.58 − 4.86i)17-s + (3.93 + 2.85i)19-s + 0.616·21-s + 6.20·23-s + (−0.809 − 0.587i)25-s + (5.10 − 15.7i)27-s + (0.154 − 0.111i)29-s + (2.49 + 7.66i)31-s + ⋯
L(s)  = 1  + (−1.54 − 1.12i)3-s + (−0.138 + 0.425i)5-s + (−0.0568 + 0.0413i)7-s + (0.822 + 2.53i)9-s + (−0.999 + 0.000485i)11-s + (0.258 + 0.794i)13-s + (0.692 − 0.503i)15-s + (0.383 − 1.18i)17-s + (0.902 + 0.655i)19-s + 0.134·21-s + 1.29·23-s + (−0.161 − 0.117i)25-s + (0.982 − 3.02i)27-s + (0.0286 − 0.0207i)29-s + (0.447 + 1.37i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.964 - 0.265i$
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: \((2,\ 440,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690220 + 0.0933582i\)
\(L(\frac12)\) \(\approx\) \(0.690220 + 0.0933582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 - 0.00161i)T \)
good3 \( 1 + (2.68 + 1.94i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (0.150 - 0.109i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.931 - 2.86i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.58 + 4.86i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.93 - 2.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.20T + 23T^{2} \)
29 \( 1 + (-0.154 + 0.111i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.49 - 7.66i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.07 + 1.50i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.24 - 5.26i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.83T + 43T^{2} \)
47 \( 1 + (-4.28 - 3.11i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.401 - 1.23i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.480 + 0.348i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.350 - 1.07i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 + (-4.03 + 12.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.723 - 0.525i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.11 - 12.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.23 - 16.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.22T + 89T^{2} \)
97 \( 1 + (-1.04 - 3.22i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.23586119303222101048945272084, −10.67232016273834703646776753460, −9.575053692911703703362326068485, −7.997001273771575917683100170506, −7.25620012150806168554666378118, −6.57498130452306134271452173389, −5.53084398367147516940847818468, −4.80097016402825234386739950481, −2.78782788511438803455866146165, −1.15515709376802519414770108579, 0.65724762114709613357131838525, 3.33886368372408694691889488432, 4.50992767880079003771678428582, 5.36296098898564492246854976559, 5.94768290629149467882922977779, 7.24912529268692229462933227225, 8.505566815609506739143725275515, 9.642890071896612645896767094934, 10.31406038234216210389328236926, 11.02397993824978626194690399928

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