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Properties

Label 1-4788-4788.3859-r0-0-0
Degree $1$
Conductor $4788$
Sign $0.534 - 0.845i$
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  + (0.766 − 0.642i)5-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.766 − 0.642i)29-s + 31-s + (0.5 + 0.866i)37-s + (0.939 + 0.342i)41-s + (0.939 + 0.342i)43-s + (−0.173 + 0.984i)47-s + (0.939 − 0.342i)53-s + (0.939 + 0.342i)55-s + (0.173 + 0.984i)59-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.766 − 0.642i)29-s + 31-s + (0.5 + 0.866i)37-s + (0.939 + 0.342i)41-s + (0.939 + 0.342i)43-s + (−0.173 + 0.984i)47-s + (0.939 − 0.342i)53-s + (0.939 + 0.342i)55-s + (0.173 + 0.984i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.534 - 0.845i$
Analytic conductor: \(22.2353\)
Root analytic conductor: \(22.2353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4788} (3859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4788,\ (0:\ ),\ 0.534 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.840300262 - 1.013727721i\)
\(L(\frac12)\) \(\approx\) \(1.840300262 - 1.013727721i\)
\(L(1)\) \(\approx\) \(1.240328407 - 0.2286722848i\)
\(L(1)\) \(\approx\) \(1.240328407 - 0.2286722848i\)

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 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 - T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−18.35215580960904905160511883761, −17.38468192828379144137765265550, −17.084852254038431037361783941201, −16.33870898247818991681648969959, −15.57283113208266326572480903525, −14.602722593029130043147866277113, −14.313110046648501640934273168, −13.73005219152115850488819508590, −12.913141786456239429367554865118, −12.1464053707397392594369432302, −11.44734862074676631089849930618, −10.77779801745661282491720311728, −10.0700459919604381655684905775, −9.44682559327838306339360823344, −8.84352486381792902291719364423, −7.85846893929018299735658469350, −7.217351987875231633199589456480, −6.41002071106945746662340970622, −5.82444517690811406555954961376, −5.23103993476923866785441795449, −4.04554865148486591844964066013, −3.50788540074012989136720833551, −2.538218659050254185644339191633, −1.89875531652650834330032961096, −0.95675837326301667688736628156, 0.63825857070401444539429910626, 1.48537643999427128621588145408, 2.38609189251420998988318393686, 2.97544062550115704888328167242, 4.388699556547978128190816930975, 4.55890861713709429153146503300, 5.66193082224752935940727045977, 6.04584093811188227768582468308, 7.07945827289137711374177594880, 7.72923571358553209775691719580, 8.45902709069888987625570972816, 9.383494568070780690181832403937, 9.8386861535305619690124730143, 10.26497087298596165077050256337, 11.40107127272304078349596184313, 12.15668676737317962332215008784, 12.56623275155276270623700882851, 13.29103008366085202167433358654, 14.058458336275131888045750422, 14.61650566579322771255462508141, 15.29377038657554371191518285097, 16.18337357101024646941665776489, 16.751250496766924798144169568288, 17.40674364147501896332354471938, 17.85340944555402721380150749990

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