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Properties

Label 2-440-440.219-c1-0-14
Degree $2$
Conductor $440$
Sign $-0.629 - 0.776i$
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  + (−0.413 + 1.35i)2-s + (−1.65 − 1.11i)4-s + 2.23i·5-s − 0.856i·7-s + (2.19 − 1.78i)8-s + 3·9-s + (−3.02 − 0.924i)10-s + 3.31·11-s + 6.26i·13-s + (1.15 + 0.353i)14-s + (1.5 + 3.70i)16-s − 6.87·17-s + (−1.23 + 4.05i)18-s + (2.49 − 3.70i)20-s + (−1.37 + 4.48i)22-s + ⋯
L(s)  = 1  + (−0.292 + 0.956i)2-s + (−0.829 − 0.559i)4-s + 0.999i·5-s − 0.323i·7-s + (0.776 − 0.629i)8-s + 9-s + (−0.956 − 0.292i)10-s + 1.00·11-s + 1.73i·13-s + (0.309 + 0.0946i)14-s + (0.375 + 0.927i)16-s − 1.66·17-s + (−0.292 + 0.956i)18-s + (0.559 − 0.829i)20-s + (−0.292 + 0.956i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.629 - 0.776i$
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: \((2,\ 440,\ (\ :1/2),\ -0.629 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481822 + 1.01057i\)
\(L(\frac12)\) \(\approx\) \(0.481822 + 1.01057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.413 - 1.35i)T \)
5 \( 1 - 2.23iT \)
11 \( 1 - 3.31T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + 0.856iT - 7T^{2} \)
13 \( 1 - 6.26iT - 13T^{2} \)
17 \( 1 + 6.87T + 17T^{2} \)
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 \( 1 - 8.52T + 43T^{2} \)
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 \( 1 + 0.261T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
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

−11.18898493914252853217935251514, −10.45627465016167566711549792309, −9.407619014247225808728072022846, −8.868030331441813475434935971963, −7.39806162339166241551887378567, −6.79181095740568489914130978115, −6.35384445473610865220369722198, −4.57323542968418820370244060559, −3.93284646746835353349176796951, −1.76684075556930808688493665619, 0.855905429141141288064684164211, 2.25872236264795694304878807226, 3.88827883951702434038043298902, 4.63986528109313714569756561661, 5.85673839051167269809535614836, 7.40180589414354876781822873532, 8.371800254716522446018279006314, 9.144962323300286900258091184565, 9.837552058400489895622650977506, 10.82270695667421849957068382850

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