Properties

Label 2-5520-92.91-c1-0-30
Degree $2$
Conductor $5520$
Sign $0.707 - 0.706i$
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 − 2.13·7-s − 9-s + 4.16·11-s + 2.81·13-s − 15-s + 1.82i·17-s − 1.32·19-s + 2.13i·21-s + (4.63 + 1.24i)23-s − 25-s + i·27-s − 4.92·29-s + 10.5i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s − 0.806·7-s − 0.333·9-s + 1.25·11-s + 0.781·13-s − 0.258·15-s + 0.441i·17-s − 0.304·19-s + 0.465i·21-s + (0.965 + 0.259i)23-s − 0.200·25-s + 0.192i·27-s − 0.914·29-s + 1.89i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.707 - 0.706i$
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: \((2,\ 5520,\ (\ :1/2),\ 0.707 - 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.408822872\)
\(L(\frac12)\) \(\approx\) \(1.408822872\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.63 - 1.24i)T \)
good7 \( 1 + 2.13T + 7T^{2} \)
11 \( 1 - 4.16T + 11T^{2} \)
13 \( 1 - 2.81T + 13T^{2} \)
17 \( 1 - 1.82iT - 17T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
29 \( 1 + 4.92T + 29T^{2} \)
31 \( 1 - 10.5iT - 31T^{2} \)
37 \( 1 - 4.95iT - 37T^{2} \)
41 \( 1 - 0.656T + 41T^{2} \)
43 \( 1 + 9.49T + 43T^{2} \)
47 \( 1 - 7.28iT - 47T^{2} \)
53 \( 1 + 7.24iT - 53T^{2} \)
59 \( 1 - 8.42iT - 59T^{2} \)
61 \( 1 - 3.54iT - 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 0.897iT - 71T^{2} \)
73 \( 1 - 4.48T + 73T^{2} \)
79 \( 1 - 7.73T + 79T^{2} \)
83 \( 1 + 3.14T + 83T^{2} \)
89 \( 1 - 5.21iT - 89T^{2} \)
97 \( 1 - 0.00964iT - 97T^{2} \)
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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

−8.427080956081786981490172623329, −7.42325112652955604432047550388, −6.64373797757489416903190882330, −6.35397418825681122708480081437, −5.47479506211421722172159873430, −4.58279056358412884270852090824, −3.61122704567370293033170037938, −3.10930747215358260252581609990, −1.70085648036335777675131237635, −1.09987119534717291911170221713, 0.40330832460303744368047953399, 1.80827743079938598770305685489, 2.92269601158444482779788967941, 3.66153165657785056074435027467, 4.14343678015387385109684625302, 5.18763512103533473551600237585, 6.08254761107419768718854467585, 6.51414430225826313392436057856, 7.23104770337914968536527151293, 8.120697373995618468918992845842

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