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Properties

Label 1-72e2-5184.83-r0-0-0
Degree $1$
Conductor $5184$
Sign $-0.259 - 0.965i$
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  + (0.653 − 0.756i)5-s + (0.474 + 0.880i)7-s + (0.900 − 0.435i)11-s + (−0.244 − 0.969i)13-s + (0.642 − 0.766i)17-s + (−0.953 + 0.300i)19-s + (0.474 − 0.880i)23-s + (−0.144 − 0.989i)25-s + (−0.999 + 0.0145i)29-s + (−0.686 + 0.727i)31-s + (0.976 + 0.216i)35-s + (−0.216 − 0.976i)37-s + (0.784 + 0.620i)41-s + (−0.827 − 0.561i)43-s + (0.727 − 0.686i)47-s + ⋯
L(s)  = 1  + (0.653 − 0.756i)5-s + (0.474 + 0.880i)7-s + (0.900 − 0.435i)11-s + (−0.244 − 0.969i)13-s + (0.642 − 0.766i)17-s + (−0.953 + 0.300i)19-s + (0.474 − 0.880i)23-s + (−0.144 − 0.989i)25-s + (−0.999 + 0.0145i)29-s + (−0.686 + 0.727i)31-s + (0.976 + 0.216i)35-s + (−0.216 − 0.976i)37-s + (0.784 + 0.620i)41-s + (−0.827 − 0.561i)43-s + (0.727 − 0.686i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ -0.259 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158445801 - 1.511611836i\)
\(L(\frac12)\) \(\approx\) \(1.158445801 - 1.511611836i\)
\(L(1)\) \(\approx\) \(1.199998317 - 0.3250720654i\)
\(L(1)\) \(\approx\) \(1.199998317 - 0.3250720654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.653 - 0.756i)T \)
7 \( 1 + (0.474 + 0.880i)T \)
11 \( 1 + (0.900 - 0.435i)T \)
13 \( 1 + (-0.244 - 0.969i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (-0.953 + 0.300i)T \)
23 \( 1 + (0.474 - 0.880i)T \)
29 \( 1 + (-0.999 + 0.0145i)T \)
31 \( 1 + (-0.686 + 0.727i)T \)
37 \( 1 + (-0.216 - 0.976i)T \)
41 \( 1 + (0.784 + 0.620i)T \)
43 \( 1 + (-0.827 - 0.561i)T \)
47 \( 1 + (0.727 - 0.686i)T \)
53 \( 1 + (0.991 + 0.130i)T \)
59 \( 1 + (-0.328 - 0.944i)T \)
61 \( 1 + (-0.159 + 0.987i)T \)
67 \( 1 + (-0.696 - 0.717i)T \)
71 \( 1 + (0.422 - 0.906i)T \)
73 \( 1 + (0.422 + 0.906i)T \)
79 \( 1 + (0.116 - 0.993i)T \)
83 \( 1 + (-0.873 - 0.487i)T \)
89 \( 1 + (0.906 - 0.422i)T \)
97 \( 1 + (-0.893 - 0.448i)T \)
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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

−18.08397506545063381279194265343, −17.34080250309421037899694230661, −16.99049495374451972931724062386, −16.539225783376204455050747712601, −15.086903647794377884143663643757, −14.9338450175760277897154806810, −14.21082216227683903662781821826, −13.647470442406404147374912971651, −12.9893667752035750252396711900, −12.11222801985565213946792118573, −11.25295412790835406168927770864, −10.92961175713967998694437350602, −10.04826442001966946785382655930, −9.51880807994060182753480099369, −8.8469613016344182531973367225, −7.795593835045549393153538357639, −7.16537585598405199404065110880, −6.65601642021282153245845641293, −5.91856881143641309528111977445, −5.07276652042322053979365433697, −4.03772874327654236631605964600, −3.804385720916685858558991832519, −2.60681025717764121021562213125, −1.74532436294552832888032636021, −1.30136291556250046389353520221, 0.468853383617880122480205300911, 1.43097154500822031310401801833, 2.15334722337179057157564773609, 2.94721831902361451685732760239, 3.89873856048040703384148982853, 4.79440365384506358526715063458, 5.47654738223337766178788018162, 5.85746459724228357054120328063, 6.75818136302210556632209794381, 7.69201524118069942891312866899, 8.49362894823452178304209127530, 8.95424484450386564282262220430, 9.50447683671519437796606747702, 10.42344789534261915218978459851, 11.03644180496601851260140639231, 12.0235546452944917949063596873, 12.37333769317272012587798471155, 13.0352711009080841056178833962, 13.81575579808508306200943929572, 14.661608246719827725235544615231, 14.87258910495690639394805593728, 15.92583549249501916809255666004, 16.60218093804189975767728537904, 17.040177716673118159655487029440, 17.797936166566826828652628891897

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