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Properties

Label 2-5472-8.5-c1-0-47
Degree $2$
Conductor $5472$
Sign $0.806 - 0.591i$
Analytic cond. $43.6941$
Root an. cond. $6.61015$
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  + 0.594i·5-s + 3.48·7-s + 4.83i·11-s + 0.215i·13-s − 1.29·17-s − i·19-s + 4.52·23-s + 4.64·25-s − 9.41i·29-s + 1.22·31-s + 2.07i·35-s − 5.62i·37-s + 0.450·41-s + 0.794i·43-s + 12.1·47-s + ⋯
L(s)  = 1  + 0.265i·5-s + 1.31·7-s + 1.45i·11-s + 0.0597i·13-s − 0.314·17-s − 0.229i·19-s + 0.944·23-s + 0.929·25-s − 1.74i·29-s + 0.219·31-s + 0.350i·35-s − 0.924i·37-s + 0.0702·41-s + 0.121i·43-s + 1.77·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5472\)    =    \(2^{5} \cdot 3^{2} \cdot 19\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(43.6941\)
Root analytic conductor: \(6.61015\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5472} (2737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5472,\ (\ :1/2),\ 0.806 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.481349435\)
\(L(\frac12)\) \(\approx\) \(2.481349435\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 - 0.594iT - 5T^{2} \)
7 \( 1 - 3.48T + 7T^{2} \)
11 \( 1 - 4.83iT - 11T^{2} \)
13 \( 1 - 0.215iT - 13T^{2} \)
17 \( 1 + 1.29T + 17T^{2} \)
23 \( 1 - 4.52T + 23T^{2} \)
29 \( 1 + 9.41iT - 29T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 + 5.62iT - 37T^{2} \)
41 \( 1 - 0.450T + 41T^{2} \)
43 \( 1 - 0.794iT - 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 2.56iT - 53T^{2} \)
59 \( 1 + 2.75iT - 59T^{2} \)
61 \( 1 - 7.76iT - 61T^{2} \)
67 \( 1 - 4.11iT - 67T^{2} \)
71 \( 1 + 7.82T + 71T^{2} \)
73 \( 1 - 3.08T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 2.08T + 97T^{2} \)
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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

−8.178463887560630219150443225836, −7.37995665690408276947568520227, −7.08594562403703934483369651288, −6.07899718624224030733741930975, −5.18677071403871324466501065410, −4.56406320055728609339074918300, −4.03852985917779768088285030253, −2.64652194249508728770173422669, −2.07664897270179556064475639009, −0.995555346271992234132656914041, 0.816889335827026370783149789324, 1.59010459265188860251965305238, 2.80595995962456670045044133977, 3.53898066248868504611671571680, 4.63015111262905664982633720144, 5.10470164943864096146505070662, 5.81873691857670437791178140318, 6.70204627714296492578261785333, 7.45006021693557081537806568592, 8.247128268717593558071640143943

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