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Properties

Label 1-5520-5520.4643-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.544 - 0.838i$
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  + (0.909 + 0.415i)7-s + (0.281 − 0.959i)11-s + (−0.415 − 0.909i)13-s + (0.540 − 0.841i)17-s + (−0.540 − 0.841i)19-s + (−0.540 + 0.841i)29-s + (0.654 − 0.755i)31-s + (−0.142 + 0.989i)37-s + (−0.142 − 0.989i)41-s + (−0.654 − 0.755i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.415 − 0.909i)53-s + (−0.909 + 0.415i)59-s + (0.755 + 0.654i)61-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)7-s + (0.281 − 0.959i)11-s + (−0.415 − 0.909i)13-s + (0.540 − 0.841i)17-s + (−0.540 − 0.841i)19-s + (−0.540 + 0.841i)29-s + (0.654 − 0.755i)31-s + (−0.142 + 0.989i)37-s + (−0.142 − 0.989i)41-s + (−0.654 − 0.755i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.415 − 0.909i)53-s + (−0.909 + 0.415i)59-s + (0.755 + 0.654i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.544 - 0.838i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.544 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6825713116 - 1.256555597i\)
\(L(\frac12)\) \(\approx\) \(0.6825713116 - 1.256555597i\)
\(L(1)\) \(\approx\) \(1.061078699 - 0.2531895328i\)
\(L(1)\) \(\approx\) \(1.061078699 - 0.2531895328i\)

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 \( 1 + (0.909 + 0.415i)T \)
11 \( 1 + (0.281 - 0.959i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (-0.540 + 0.841i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (-0.142 + 0.989i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (-0.909 + 0.415i)T \)
61 \( 1 + (0.755 + 0.654i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.989 - 0.142i)T \)
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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

−18.01970609332689684765404901711, −17.25551432037590629435230863881, −17.00799297626711659149071685977, −16.23180680046209304702126035604, −15.29251955095232878258923219793, −14.645462962845894807557082398513, −14.352989549939063209071174198800, −13.57103801953437641351565656374, −12.60646704812462890268129233103, −12.20485715976550389113061909080, −11.42805262880785567637721246000, −10.79614292762357442106629890359, −9.99456747117183514247499927582, −9.5346919426188044226138211960, −8.54494247740207201459544674450, −7.94565373926243595954571988712, −7.33096087257104012200142243168, −6.56776742475631353032131283540, −5.841831549147746486760001776708, −4.88078638131826317430524420392, −4.31318317254932938197750443383, −3.78872959945305800860907576090, −2.59421079318039152310156468859, −1.71719929623240925835375066948, −1.33458250718625584319274077500, 0.359771800998544861994104993106, 1.25643290531962456248018276304, 2.25453610207655077825974184376, 2.94597461207445969284936342671, 3.7051831701685437201048874597, 4.73861358249650652413633088905, 5.282069742252417071240361141733, 5.86062058697500164631669778498, 6.84815021932713592014653590435, 7.503989614411960477739906234544, 8.33644583657669071719627853454, 8.72111695219982482625888840105, 9.544943939561054350438347042307, 10.42351988246259100635939544441, 10.954834607003169927287905199396, 11.82802209296303348134069684862, 12.02911286882745320338331508928, 13.19973432909700755800501505465, 13.59445633343196561751415361742, 14.447583983012843932501973750715, 15.00542885099639416835266871313, 15.53712441152615627422371595634, 16.39433127301998461794235625563, 17.067014425403333396767895408037, 17.57242889690890249589332175108

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