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Properties

Label 1-4788-4788.151-r0-0-0
Degree $1$
Conductor $4788$
Sign $0.281 - 0.959i$
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  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s − 47-s + (0.5 + 0.866i)53-s − 55-s + 59-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s − 47-s + (0.5 + 0.866i)53-s − 55-s + 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(22.2353\)
Root analytic conductor: \(22.2353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4788} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4788,\ (0:\ ),\ 0.281 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448991576 - 1.084598277i\)
\(L(\frac12)\) \(\approx\) \(1.448991576 - 1.084598277i\)
\(L(1)\) \(\approx\) \(1.056225113 - 0.3155438600i\)
\(L(1)\) \(\approx\) \(1.056225113 - 0.3155438600i\)

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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−18.26429319157804906453289429846, −17.64674765879355452216296864700, −17.01366167263622261325822707315, −16.196810002982295168118083771700, −15.50948748833046513106836299356, −14.877043769654624581035338420550, −14.44822927104264348363578840954, −13.63000850667943568554576966536, −12.903029924399059226343056916665, −12.067220926572024809728647287214, −11.508682125793540906631991897745, −10.93343810612388308269290472398, −10.091360496925783222160496652354, −9.602978869679425891510312492460, −8.50119912129188714453096427872, −8.14483193589781737843108711697, −7.03543792995574493117550128184, −6.62770835669395733640967595384, −6.11157124114791213599132857457, −4.80077506232145936980492677989, −4.213315553890158188257907647047, −3.62995098602496930754194462978, −2.57334717170979796720779496881, −2.00166154499662285927279902823, −0.87438909054374818896163427793, 0.71543932510988740305780770212, 1.10076876146008027248274029500, 2.393775072032733992040072615, 3.31392871127225798519189784136, 3.87984745872824997942843476428, 4.81760710424387291121041268577, 5.39600482403244314580611115510, 6.15989996983512402572886458197, 7.03395294116169246734420313898, 7.836821293208230759384364170702, 8.44819575281157110310160374291, 9.06965561882906957850307983017, 9.64906175875074377195856520474, 10.75428927595970662029637827274, 11.262098796056945407554634108666, 11.904133545183266172747021780348, 12.654307824581490586629241303683, 13.29352472275048336862091337042, 13.84532445821968123845308856689, 14.68495742350186749969428496879, 15.53908985440606860428503893797, 16.041274039355374948845473318113, 16.47045582447396504766212423076, 17.43911689345602355115565863287, 17.813174658138378801896054572886

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