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Properties

Label 1-72e2-5184.59-r0-0-0
Degree $1$
Conductor $5184$
Sign $-0.421 - 0.906i$
Analytic cond. $24.0743$
Root an. cond. $24.0743$
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  + (−0.987 − 0.159i)5-s + (−0.144 + 0.989i)7-s + (−0.409 − 0.912i)11-s + (0.997 + 0.0726i)13-s + (0.342 − 0.939i)17-s + (−0.737 + 0.675i)19-s + (−0.144 − 0.989i)23-s + (0.949 + 0.314i)25-s + (0.187 − 0.982i)29-s + (0.396 + 0.918i)31-s + (0.300 − 0.953i)35-s + (−0.953 + 0.300i)37-s + (0.747 + 0.664i)41-s + (0.994 + 0.101i)43-s + (−0.918 − 0.396i)47-s + ⋯
L(s)  = 1  + (−0.987 − 0.159i)5-s + (−0.144 + 0.989i)7-s + (−0.409 − 0.912i)11-s + (0.997 + 0.0726i)13-s + (0.342 − 0.939i)17-s + (−0.737 + 0.675i)19-s + (−0.144 − 0.989i)23-s + (0.949 + 0.314i)25-s + (0.187 − 0.982i)29-s + (0.396 + 0.918i)31-s + (0.300 − 0.953i)35-s + (−0.953 + 0.300i)37-s + (0.747 + 0.664i)41-s + (0.994 + 0.101i)43-s + (−0.918 − 0.396i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.421 - 0.906i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ -0.421 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3690207419 - 0.5784726276i\)
\(L(\frac12)\) \(\approx\) \(0.3690207419 - 0.5784726276i\)
\(L(1)\) \(\approx\) \(0.8018267895 - 0.05524911000i\)
\(L(1)\) \(\approx\) \(0.8018267895 - 0.05524911000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.987 - 0.159i)T \)
7 \( 1 + (-0.144 + 0.989i)T \)
11 \( 1 + (-0.409 - 0.912i)T \)
13 \( 1 + (0.997 + 0.0726i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-0.737 + 0.675i)T \)
23 \( 1 + (-0.144 - 0.989i)T \)
29 \( 1 + (0.187 - 0.982i)T \)
31 \( 1 + (0.396 + 0.918i)T \)
37 \( 1 + (-0.953 + 0.300i)T \)
41 \( 1 + (0.747 + 0.664i)T \)
43 \( 1 + (0.994 + 0.101i)T \)
47 \( 1 + (-0.918 - 0.396i)T \)
53 \( 1 + (-0.991 + 0.130i)T \)
59 \( 1 + (-0.355 - 0.934i)T \)
61 \( 1 + (-0.487 - 0.873i)T \)
67 \( 1 + (0.827 + 0.561i)T \)
71 \( 1 + (0.573 - 0.819i)T \)
73 \( 1 + (0.573 + 0.819i)T \)
79 \( 1 + (0.998 + 0.0581i)T \)
83 \( 1 + (-0.328 - 0.944i)T \)
89 \( 1 + (-0.819 + 0.573i)T \)
97 \( 1 + (-0.973 - 0.230i)T \)
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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

−18.05422544013110325491060193497, −17.60178561135106168966661216209, −16.84052429915662763881087950006, −16.155199645990560132105474561780, −15.45433181629397476733596435186, −15.111435886022377884686164711036, −14.18201305016059798030004065053, −13.56501004350196639404901987549, −12.653025574053935857715376832613, −12.460048895776656791924393565503, −11.22273060766732618149575824970, −10.93095991919879490946399320785, −10.29109651754216189228991866790, −9.461448258775964342227634391640, −8.570979175993389487086567188338, −7.91280295262134194399354648894, −7.337656054910205450091186205413, −6.73188887268270538844978000980, −5.9144192573445330275271614669, −4.8960601702983491366051809842, −4.14309605060620900880233024517, −3.73410807885548829018925631291, −2.91577413767785421302996515495, −1.79486138242571069232997205066, −0.94930143611997137547473638528, 0.222486725383357902823421274514, 1.212267884914057492901563520040, 2.36937604963676405172345224017, 3.114274869554449636423278340281, 3.70562536040462949750485909911, 4.64509683455264301165695790726, 5.30051992604761853680557894634, 6.21654131154264095314010039309, 6.61470166670821297310915979685, 7.89552888344281633042354417467, 8.21802142508283102672259043944, 8.79137693827133662867906384174, 9.56195645477349443795698633576, 10.548196496935918926383494285932, 11.16328191143954338135086809648, 11.72390201882254455167636471433, 12.46196774241806477405045936978, 12.87905510086713702724255567215, 13.93703755185590066346369509371, 14.38358018174693976589151620386, 15.39234621997919146380185741317, 15.798029696288914526528159116253, 16.247764529880592837160772143609, 16.9184945136698102722920101379, 18.02358070327038745004231265796

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