↑

Properties

Label 1-5520-5520.4397-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.205 + 0.978i$
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  + (0.281 + 0.959i)7-s + (0.755 − 0.654i)11-s + (−0.959 − 0.281i)13-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (−0.989 − 0.142i)29-s + (0.841 + 0.540i)31-s + (0.415 − 0.909i)37-s + (0.415 + 0.909i)41-s + (−0.841 + 0.540i)43-s − i·47-s + (−0.841 + 0.540i)49-s + (−0.959 + 0.281i)53-s + (−0.281 + 0.959i)59-s + (−0.540 + 0.841i)61-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)7-s + (0.755 − 0.654i)11-s + (−0.959 − 0.281i)13-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (−0.989 − 0.142i)29-s + (0.841 + 0.540i)31-s + (0.415 − 0.909i)37-s + (0.415 + 0.909i)41-s + (−0.841 + 0.540i)43-s − i·47-s + (−0.841 + 0.540i)49-s + (−0.959 + 0.281i)53-s + (−0.281 + 0.959i)59-s + (−0.540 + 0.841i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.205 + 0.978i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.205 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8115014221 + 0.9998456408i\)
\(L(\frac12)\) \(\approx\) \(0.8115014221 + 0.9998456408i\)
\(L(1)\) \(\approx\) \(1.006604461 + 0.1709434785i\)
\(L(1)\) \(\approx\) \(1.006604461 + 0.1709434785i\)

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

−17.30010428175723049189729045481, −17.14279339596663841331490662743, −16.66075466634092816335713947101, −15.641035805180768341292018246169, −14.89101007968863655693772057604, −14.40755318711292372816336485618, −13.87894367452976927189411921171, −12.96579423703008129411920535651, −12.40783885415789691453897128189, −11.68032544757168664661689065514, −11.057707416888610748287751004614, −10.19548655004582460432491367980, −9.74121444594193688156471325478, −9.06890980156392930241671819774, −8.015673637303708795653487020314, −7.5744202310612899477689852659, −6.81548169366337321554010318310, −6.26400195033422743296833389715, −5.15199979661789054191214372127, −4.55026270193241165623195790984, −3.9483533250553331623834214040, −3.117264483646116017862502653597, −2.06751367923929282567811174803, −1.44557160034685231945018454838, −0.350696897325340516260067063593, 1.013913567269440932552774828476, 1.90505761631685925328151411566, 2.68350735419115597718535572364, 3.41008797375656827232031897173, 4.30122594930928562846182192588, 5.08841303054721832861253298823, 5.80289663648237022861501190267, 6.326323468601715398246006664789, 7.249683587888975225782326566935, 8.0839805434540451592016131528, 8.53276049518418217185515486187, 9.40446986992291253904252042918, 9.8614410573484892745120583268, 10.84061378882690076949478579751, 11.41556945088539446640997443727, 12.25074685322353042545748004448, 12.47747858001043415978879978315, 13.44338864478861448942433682347, 14.257264601858304410148356034751, 14.87446521808544734555551121592, 15.14453130221422308387152598717, 16.20664880995027510526178694609, 16.77486529750280113136630111433, 17.28642014419126864617947058438, 18.15755928798811153302745084084

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