Properties

Label 1-5520-5520.3893-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

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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: $\chi_{5520} (3893, \cdot )$
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 \( 1 + (-0.909 - 0.415i)T \)
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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

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

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