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Properties

Label 1-4788-4788.263-r0-0-0
Degree $1$
Conductor $4788$
Sign $-0.220 - 0.975i$
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  + (−0.173 − 0.984i)5-s + 11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.173 − 0.984i)41-s + (−0.173 − 0.984i)43-s + (0.173 − 0.984i)47-s + (−0.766 − 0.642i)53-s + (−0.173 − 0.984i)55-s + (0.173 + 0.984i)59-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)5-s + 11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.173 − 0.984i)41-s + (−0.173 − 0.984i)43-s + (0.173 − 0.984i)47-s + (−0.766 − 0.642i)53-s + (−0.173 − 0.984i)55-s + (0.173 + 0.984i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(22.2353\)
Root analytic conductor: \(22.2353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4788} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4788,\ (0:\ ),\ -0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7808910969 - 0.9775566782i\)
\(L(\frac12)\) \(\approx\) \(0.7808910969 - 0.9775566782i\)
\(L(1)\) \(\approx\) \(0.9535032960 - 0.2210303962i\)
\(L(1)\) \(\approx\) \(0.9535032960 - 0.2210303962i\)

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

−18.30291101535430927975208138313, −17.576119553643375910411347672441, −17.24323978247582393164062029991, −16.181094714596450083828936458276, −15.68557937396480113653181287591, −14.770447632135546664331606537601, −14.46004422977097537257287509795, −13.83718683842958942740007044699, −12.90071194558091790997732588230, −12.16656796425592218774316463515, −11.58555842817697177228662157783, −10.85121693128959252536385602789, −10.30509531043213423330157444830, −9.45411103443051521599990439510, −8.900405822300664836401202045468, −7.91901400059135444177100506711, −7.24633648541823504609534300525, −6.59059322183327087887681558628, −6.15651492982054411950508803689, −4.84357274000192459638722183717, −4.51897449304096245011634394774, −3.30447451469377715572902096727, −2.89467938228427361601415302447, −2.02566790383017343715892358980, −0.93880412567476434550987755056, 0.38268154557187263244703166843, 1.43330104384820717264330263874, 2.06966141153926927414779367597, 3.167394684339550724800195414631, 4.1235743396812216746253921174, 4.54737534492198192629246212038, 5.34639280912516952658902335744, 6.17424057886167971288071620096, 6.9556572922719654690128000213, 7.6106527528440745149928422552, 8.71578501331337987305631336237, 8.8108145903998991841603043633, 9.75264767272293714840750866048, 10.347393819029587185469700765629, 11.47080214119974113286315440179, 11.88590180884506096573924164466, 12.43978272332404760687366364871, 13.351955651526715009915781836073, 13.72767938560548730067060969772, 14.74099185411418018054654139982, 15.26582241302477628004725917739, 15.93723755936062090500852620676, 16.83266347159974741943025597754, 17.20657665465286148378501406612, 17.59902700014026976887443354103

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