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Properties

Label 1-4788-4788.167-r0-0-0
Degree $1$
Conductor $4788$
Sign $0.799 - 0.600i$
Analytic cond. $22.2353$
Root an. cond. $22.2353$
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  + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)23-s + (0.766 − 0.642i)25-s + (0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s − 37-s + (−0.766 − 0.642i)41-s + (−0.766 − 0.642i)43-s + (−0.173 − 0.984i)47-s + (0.173 + 0.984i)53-s + (0.766 + 0.642i)55-s + (0.173 − 0.984i)59-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)23-s + (0.766 − 0.642i)25-s + (0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s − 37-s + (−0.766 − 0.642i)41-s + (−0.766 − 0.642i)43-s + (−0.173 − 0.984i)47-s + (0.173 + 0.984i)53-s + (0.766 + 0.642i)55-s + (0.173 − 0.984i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.799 - 0.600i$
Analytic conductor: \(22.2353\)
Root analytic conductor: \(22.2353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4788} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4788,\ (0:\ ),\ 0.799 - 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.091036208 - 0.3639846525i\)
\(L(\frac12)\) \(\approx\) \(1.091036208 - 0.3639846525i\)
\(L(1)\) \(\approx\) \(0.8793461161 + 0.0002414978159i\)
\(L(1)\) \(\approx\) \(0.8793461161 + 0.0002414978159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 - T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−18.12759786407079360870040288289, −17.61143702779031627667330122199, −16.82439155657339055161872875325, −16.09522576534345504009862791455, −15.46378692371294276092377635901, −15.11378628101365260822564214721, −14.27548679061196302148008814443, −13.30917863968216488700645257408, −12.85161000975356919534363618256, −12.0620375210324441026061566477, −11.63412690271535680565294626804, −10.74550532005356422265363950608, −10.05943856718139385365444219914, −9.42948634029551052403569390763, −8.41409472249960185648823937325, −7.97314047535431201335011950642, −7.302065173651624976263140120874, −6.64812882909744884280377511305, −5.430415636260562089015783500337, −5.03547543285540581272329404227, −4.26027180844523563689720480639, −3.28117452634521544998298844104, −2.86049664323189588733099074502, −1.589146602104311538181156957568, −0.74652602632770709434807527978, 0.44249676292868235305655262990, 1.50442624475236875392841838539, 2.54454950113729207344396425663, 3.40163355786218488637560680744, 3.84245262785005549213612909200, 4.80510381873223062570396599993, 5.505192943617638005397152976072, 6.4987940050319025837519349111, 6.9885406302501996840306802176, 7.829718959487956920676857331983, 8.53157714342168774034890981536, 8.93419166864869992631343419566, 10.164945581462101166915620403136, 10.61897913214920830313069120255, 11.354667920567501417730928596581, 11.93472908940133846205203817826, 12.57347546593483047033445736142, 13.44979382556448693326164784036, 14.083981521464740115203488433525, 14.79632733174655828415270346497, 15.36754513490902561942331449704, 16.13729588062797079407963924647, 16.62017032282808825044789413672, 17.24113776655651605883208339409, 18.36260545990588334475211363152

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