Properties

Label 2-5520-92.91-c1-0-59
Degree $2$
Conductor $5520$
Sign $0.991 - 0.128i$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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  + i·3-s + i·5-s + 1.44·7-s − 9-s − 0.540·11-s + 4.25·13-s − 15-s − 7.14i·17-s − 0.941·19-s + 1.44i·21-s + (2.91 + 3.80i)23-s − 25-s i·27-s + 1.26·29-s − 3.15i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s + 0.547·7-s − 0.333·9-s − 0.162·11-s + 1.18·13-s − 0.258·15-s − 1.73i·17-s − 0.215·19-s + 0.315i·21-s + (0.607 + 0.794i)23-s − 0.200·25-s − 0.192i·27-s + 0.234·29-s − 0.566i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.167857160\)
\(L(\frac12)\) \(\approx\) \(2.167857160\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 - iT \)
23 \( 1 + (-2.91 - 3.80i)T \)
good7 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 + 0.540T + 11T^{2} \)
13 \( 1 - 4.25T + 13T^{2} \)
17 \( 1 + 7.14iT - 17T^{2} \)
19 \( 1 + 0.941T + 19T^{2} \)
29 \( 1 - 1.26T + 29T^{2} \)
31 \( 1 + 3.15iT - 31T^{2} \)
37 \( 1 - 1.30iT - 37T^{2} \)
41 \( 1 - 5.97T + 41T^{2} \)
43 \( 1 - 2.03T + 43T^{2} \)
47 \( 1 + 6.61iT - 47T^{2} \)
53 \( 1 + 7.54iT - 53T^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 - 2.00iT - 61T^{2} \)
67 \( 1 - 7.68T + 67T^{2} \)
71 \( 1 - 8.07iT - 71T^{2} \)
73 \( 1 - 1.91T + 73T^{2} \)
79 \( 1 + 3.99T + 79T^{2} \)
83 \( 1 - 1.20T + 83T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + 11.5iT - 97T^{2} \)
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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

−8.227823882928784488108894537803, −7.45625506043271247673860075366, −6.79090322636573731023022989511, −5.92781528720606260276266621261, −5.21566347570627929929096028356, −4.56234961444175527639208448720, −3.63788711825347635478031257956, −2.98461610579503910409299717453, −1.98698648668124735995178515401, −0.69442807778164191278251439408, 0.982688731578898687587939757574, 1.65139694630035142733947691992, 2.67845830509998959074600869524, 3.76098871383341900621376180316, 4.43278872652478236929969024743, 5.33361096303703874823121421061, 6.13845150402055229620231123657, 6.52669851100108099697245989598, 7.64018167724224639118380063176, 8.118408748424025531666690550848

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