Properties

Label 2-5520-92.91-c1-0-93
Degree $2$
Conductor $5520$
Sign $-0.949 + 0.314i$
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 + 0.482·7-s − 9-s + 1.10·11-s + 2.25·13-s + 15-s − 6.03i·17-s − 1.48·19-s + 0.482i·21-s + (−4.55 + 1.50i)23-s − 25-s i·27-s − 7.68·29-s − 5.12i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s + 0.182·7-s − 0.333·9-s + 0.332·11-s + 0.625·13-s + 0.258·15-s − 1.46i·17-s − 0.340·19-s + 0.105i·21-s + (−0.949 + 0.314i)23-s − 0.200·25-s − 0.192i·27-s − 1.42·29-s − 0.920i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.949 + 0.314i$
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: \((2,\ 5520,\ (\ :1/2),\ -0.949 + 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2131853111\)
\(L(\frac12)\) \(\approx\) \(0.2131853111\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.55 - 1.50i)T \)
good7 \( 1 - 0.482T + 7T^{2} \)
11 \( 1 - 1.10T + 11T^{2} \)
13 \( 1 - 2.25T + 13T^{2} \)
17 \( 1 + 6.03iT - 17T^{2} \)
19 \( 1 + 1.48T + 19T^{2} \)
29 \( 1 + 7.68T + 29T^{2} \)
31 \( 1 + 5.12iT - 31T^{2} \)
37 \( 1 + 5.83iT - 37T^{2} \)
41 \( 1 - 1.27T + 41T^{2} \)
43 \( 1 + 8.46T + 43T^{2} \)
47 \( 1 - 11.2iT - 47T^{2} \)
53 \( 1 - 8.81iT - 53T^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 - 0.0392iT - 61T^{2} \)
67 \( 1 + 8.39T + 67T^{2} \)
71 \( 1 + 1.25iT - 71T^{2} \)
73 \( 1 + 3.47T + 73T^{2} \)
79 \( 1 + 3.61T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 - 6.85iT - 89T^{2} \)
97 \( 1 - 0.330iT - 97T^{2} \)
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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

−7.82279921134362130049230726095, −7.29410359339321618128211574000, −6.17315024142316806061906659111, −5.70267669642701683211463882812, −4.81668216406091667762208432680, −4.19101514025169836103319343886, −3.45829118832373087136542538665, −2.43752139127375606433766398807, −1.37547971406580879421038369758, −0.05285082723860606414666445949, 1.53669665684338924185142910211, 2.03716071774078843012859783530, 3.36092782645457952142397527259, 3.81905953280320393686400620959, 4.89790059210488864761131761196, 5.81992270800462337695016594905, 6.41634663384842635580742653490, 6.90075889225919445801753123346, 7.88122845321434871269697989703, 8.345759361748783435456820387202

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