Properties

Label 2-5520-92.91-c1-0-35
Degree $2$
Conductor $5520$
Sign $0.918 - 0.395i$
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 − 1.61·7-s − 9-s − 1.15·11-s + 7.10·13-s − 15-s + 1.79i·17-s − 5.93·19-s + 1.61i·21-s + (−4.40 + 1.89i)23-s − 25-s + i·27-s − 0.0365·29-s − 4.39i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s − 0.609·7-s − 0.333·9-s − 0.346·11-s + 1.97·13-s − 0.258·15-s + 0.434i·17-s − 1.36·19-s + 0.351i·21-s + (−0.918 + 0.395i)23-s − 0.200·25-s + 0.192i·27-s − 0.00679·29-s − 0.789i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.918 - 0.395i$
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: \((2,\ 5520,\ (\ :1/2),\ 0.918 - 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.386596061\)
\(L(\frac12)\) \(\approx\) \(1.386596061\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.40 - 1.89i)T \)
good7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 + 1.15T + 11T^{2} \)
13 \( 1 - 7.10T + 13T^{2} \)
17 \( 1 - 1.79iT - 17T^{2} \)
19 \( 1 + 5.93T + 19T^{2} \)
29 \( 1 + 0.0365T + 29T^{2} \)
31 \( 1 + 4.39iT - 31T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 - 5.43T + 41T^{2} \)
43 \( 1 - 9.60T + 43T^{2} \)
47 \( 1 - 9.16iT - 47T^{2} \)
53 \( 1 - 1.95iT - 53T^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 + 9.53iT - 61T^{2} \)
67 \( 1 - 2.33T + 67T^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + 1.61T + 73T^{2} \)
79 \( 1 - 5.33T + 79T^{2} \)
83 \( 1 - 0.722T + 83T^{2} \)
89 \( 1 + 2.09iT - 89T^{2} \)
97 \( 1 - 1.93iT - 97T^{2} \)
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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

−8.159589940125641535048185236365, −7.69411126460975596614827383374, −6.42392643703815654251523025035, −6.26856177571455666790412198142, −5.58191582236128222026512829703, −4.36067683048770041820642940750, −3.81299997677443696634781872084, −2.83074598331222616019293132557, −1.81804114457543391763559866390, −0.909025478744700049156887050374, 0.43770958148564349140230039713, 1.95935012065823213345418196377, 2.87437352652308085635460015467, 3.79186262729980757578816117301, 4.13204587671313374324146393152, 5.30496117654228583775571422427, 6.11671998804762450915493567622, 6.42408204759593232073086205774, 7.39652714133598161690702626691, 8.258557308168757794301672951866

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