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Properties

Label 1-5520-5520.2933-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.551 - 0.834i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.989 − 0.142i)7-s + (0.909 − 0.415i)11-s + (−0.142 − 0.989i)13-s + (−0.755 − 0.654i)17-s + (0.755 − 0.654i)19-s + (0.755 + 0.654i)29-s + (−0.959 + 0.281i)31-s + (0.841 + 0.540i)37-s + (0.841 − 0.540i)41-s + (0.959 + 0.281i)43-s + i·47-s + (0.959 + 0.281i)49-s + (−0.142 + 0.989i)53-s + (0.989 − 0.142i)59-s + (0.281 + 0.959i)61-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)7-s + (0.909 − 0.415i)11-s + (−0.142 − 0.989i)13-s + (−0.755 − 0.654i)17-s + (0.755 − 0.654i)19-s + (0.755 + 0.654i)29-s + (−0.959 + 0.281i)31-s + (0.841 + 0.540i)37-s + (0.841 − 0.540i)41-s + (0.959 + 0.281i)43-s + i·47-s + (0.959 + 0.281i)49-s + (−0.142 + 0.989i)53-s + (0.989 − 0.142i)59-s + (0.281 + 0.959i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.551 - 0.834i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2933, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.551 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375360401 - 0.7395809613i\)
\(L(\frac12)\) \(\approx\) \(1.375360401 - 0.7395809613i\)
\(L(1)\) \(\approx\) \(1.008615583 - 0.1572099921i\)
\(L(1)\) \(\approx\) \(1.008615583 - 0.1572099921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.989 - 0.142i)T \)
11 \( 1 + (0.909 - 0.415i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (0.755 - 0.654i)T \)
29 \( 1 + (0.755 + 0.654i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (0.841 + 0.540i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (0.989 - 0.142i)T \)
61 \( 1 + (0.281 + 0.959i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.92458847574561097717292337288, −17.29297212954274486055558351083, −16.521639591507225684973360143244, −16.153757442986877389760104567655, −15.372050225786464363159781082804, −14.579304966532990090008957200343, −14.13705371921698428402916960286, −13.22186071411134199434028416676, −12.73299539345612774872051821239, −11.9319555860108555430207832301, −11.49696277359498923642340509890, −10.56478647725670130467505442921, −9.77112997944076415688479652155, −9.318462821991579315722434215422, −8.74548500491697388468828240059, −7.74345064715514805831803245933, −7.01528421089210672291822486143, −6.372759908060267582239679610385, −5.90149868062986256286295576291, −4.83579164951858592194549768030, −3.970397369366241106051682455318, −3.63566672651398842759581203554, −2.453588251789907089536103803324, −1.87941124104439837881204764177, −0.79568433162371149572489666587, 0.56061065870574539666325549185, 1.25141747694572157377597848827, 2.68364944870095674901648137049, 2.96379400390836577886475744793, 3.90148523811159477673067008897, 4.61160601952069432600630609388, 5.5892779902496180095477718034, 6.12028633983893317507866053143, 7.01366627077731770773834975164, 7.366765645356500168626329131048, 8.45425911572640019032276737450, 9.16462989140596075766901752975, 9.56386984321869218767691760058, 10.44076773505552747819407656110, 11.08875445505791002604254419650, 11.76519043805024090019187168785, 12.602255812705966417043207828924, 13.04652374339745748527966101882, 13.82642207529658178136851656756, 14.3625478515053487300123902424, 15.224814510673889048656429015109, 15.980169281959861664171076543341, 16.22960960527165922078059934013, 17.15537701843018175688001471984, 17.763712311251138402509242407234

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