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Properties

Label 2-440-440.219-c1-0-34
Degree $2$
Conductor $440$
Sign $0.776 - 0.629i$
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  + (1.35 + 0.413i)2-s + (1.65 + 1.11i)4-s − 2.23i·5-s + 5.22i·7-s + (1.78 + 2.19i)8-s + 3·9-s + (0.924 − 3.02i)10-s − 3.31·11-s − 3.56i·13-s + (−2.15 + 7.06i)14-s + (1.5 + 3.70i)16-s + 4.55·17-s + (4.05 + 1.23i)18-s + (2.49 − 3.70i)20-s + (−4.48 − 1.37i)22-s + ⋯
L(s)  = 1  + (0.956 + 0.292i)2-s + (0.829 + 0.559i)4-s − 0.999i·5-s + 1.97i·7-s + (0.629 + 0.776i)8-s + 9-s + (0.292 − 0.956i)10-s − 1.00·11-s − 0.989i·13-s + (−0.576 + 1.88i)14-s + (0.375 + 0.927i)16-s + 1.10·17-s + (0.956 + 0.292i)18-s + (0.559 − 0.829i)20-s + (−0.956 − 0.292i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.776 - 0.629i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 0.776 - 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39340 + 0.847976i\)
\(L(\frac12)\) \(\approx\) \(2.39340 + 0.847976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.413i)T \)
5 \( 1 + 2.23iT \)
11 \( 1 + 3.31T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 5.22iT - 7T^{2} \)
13 \( 1 + 3.56iT - 13T^{2} \)
17 \( 1 - 4.55T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.94iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 9.96T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + 17.0T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.72992776805557076840855379139, −10.34942729975050510635705350414, −9.358582619043435355250669694224, −8.193690244787279364667765282787, −7.72534379608498835510152759457, −6.07306141971815345200993968253, −5.42983292874079523908372954729, −4.75878713840564443509196778712, −3.21652829301743973662760598595, −2.00221807795488904148558950136, 1.51273901426729815649090301606, 3.20377256278119633203474158549, 4.02597956323174448780330672281, 4.98660225949117995124282830350, 6.52709959235934808021726601013, 7.13880551671549791550874448004, 7.72604409181563599846670677217, 9.929923696670362365153670149085, 10.29328640366246344961263581096, 10.89924304149410840665522379671

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