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Properties

Label 2-440-440.219-c1-0-32
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

\(L(1)\) \(\approx\) \(1.48846 + 0.709671i\)
\(L(\frac12)\) \(\approx\) \(1.48846 + 0.709671i\)
\(L(\frac{3}{2})\) 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

−11.66208518182085144817021675331, −9.819147569317204792552699914042, −9.438715897276313416837001590212, −8.428345013739192940105762745825, −7.44639975518371170965584024142, −6.65683359409199541009275770173, −5.57196517466677408526459482580, −4.37235030950362240854056711885, −3.91418779930792673463398559075, −1.34942630710739335362661198599, 1.37218370888045847559374843920, 3.00352123287499253346728047560, 3.68880475980048710200765409470, 5.13541439739446968252117506256, 6.10923082118420682285931258375, 7.30194273942333045168123277657, 8.368639861695920889153780865430, 9.664812337651073746244426914471, 10.21865846513264841726896406894, 10.86237800021760056071852307632

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