↑

Properties

Label 1-5520-5520.3659-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.344 - 0.938i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.344 - 0.938i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (3659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.344 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4166977613 - 0.5971048379i\)
\(L(\frac12)\) \(\approx\) \(0.4166977613 - 0.5971048379i\)
\(L(1)\) \(\approx\) \(0.9234210641 + 0.007779841252i\)
\(L(1)\) \(\approx\) \(0.9234210641 + 0.007779841252i\)

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

Imaginary part of the first few zeros on the critical line

−17.785089407606730974792806953481, −17.39885213695144459195842961099, −16.77875489284358100816926041037, −16.33100274622533009134828495174, −15.31507299270566769348431571094, −14.689738267080283796780181717071, −14.143439484808012848452989006401, −13.55168342056546179906372089310, −12.727399809931481958049396334836, −12.139331839247986491259624960444, −11.360588336616039800900724031430, −10.71144755167419447251848608185, −10.08131712447403181692044839911, −9.45366753031994940681334078899, −8.479563822068341059183005658123, −8.08647517709815585940955648338, −7.06798619400586213730161721772, −6.60882683865716830582534491946, −5.93144665600024388555315552236, −4.79018194568292154341289142644, −4.32145434145289254927860551878, −3.62871379779798355304631308191, −2.74074262327825951258195197459, −1.73997557333176023996957388395, −1.06683445615162940417847942153, 0.19064211091225205353623826498, 1.43621318201502468351614446995, 2.43194942778876563988084564924, 2.68099956840845080263419326804, 3.92732115085164093314753707122, 4.69083773343301417512591609927, 5.16000465817422341914518397881, 6.19177364371098437111014819546, 6.712026791511333062366851222654, 7.445025745148911715580852966966, 8.34560470583560120731816306134, 8.98704644928335792081060250021, 9.49395214316763261366742566093, 10.23189965282954400337401821084, 11.116595198050177863792473951648, 11.86668330324633312131892387734, 12.17329411723484760107551615519, 12.946734157523280709840205769, 13.77382358644025372507793969470, 14.44746341488354408473673551616, 15.12697168601784873499779425837, 15.511486409637639860998603469122, 16.329099982088128262851009144489, 17.15731742894379316015069415096, 17.68585867514870508935487852718

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