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Properties

Label 1-4851-4851.1987-r0-0-0
Degree $1$
Conductor $4851$
Sign $0.845 - 0.533i$
Analytic cond. $22.5279$
Root an. cond. $22.5279$
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  + (0.163 + 0.986i)2-s + (−0.946 + 0.323i)4-s + (0.280 + 0.959i)5-s + (−0.473 − 0.880i)8-s + (−0.900 + 0.433i)10-s + (−0.772 + 0.635i)13-s + (0.791 − 0.611i)16-s + (0.858 + 0.512i)17-s + (−0.809 − 0.587i)19-s + (−0.575 − 0.817i)20-s + (0.955 + 0.294i)23-s + (−0.842 + 0.538i)25-s + (−0.753 − 0.657i)26-s + (0.599 − 0.800i)29-s + (0.978 + 0.207i)31-s + (0.733 + 0.680i)32-s + ⋯
L(s)  = 1  + (0.163 + 0.986i)2-s + (−0.946 + 0.323i)4-s + (0.280 + 0.959i)5-s + (−0.473 − 0.880i)8-s + (−0.900 + 0.433i)10-s + (−0.772 + 0.635i)13-s + (0.791 − 0.611i)16-s + (0.858 + 0.512i)17-s + (−0.809 − 0.587i)19-s + (−0.575 − 0.817i)20-s + (0.955 + 0.294i)23-s + (−0.842 + 0.538i)25-s + (−0.753 − 0.657i)26-s + (0.599 − 0.800i)29-s + (0.978 + 0.207i)31-s + (0.733 + 0.680i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(22.5279\)
Root analytic conductor: \(22.5279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (1987, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (0:\ ),\ 0.845 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4261154097 - 0.1231082263i\)
\(L(\frac12)\) \(\approx\) \(0.4261154097 - 0.1231082263i\)
\(L(1)\) \(\approx\) \(0.7231337675 + 0.5289511696i\)
\(L(1)\) \(\approx\) \(0.7231337675 + 0.5289511696i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.163 + 0.986i)T \)
5 \( 1 + (0.280 + 0.959i)T \)
13 \( 1 + (-0.772 + 0.635i)T \)
17 \( 1 + (0.858 + 0.512i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (0.599 - 0.800i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.393 - 0.919i)T \)
41 \( 1 + (-0.999 - 0.0299i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (-0.712 - 0.701i)T \)
53 \( 1 + (0.858 - 0.512i)T \)
59 \( 1 + (-0.525 - 0.850i)T \)
61 \( 1 + (0.0149 - 0.999i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.995 - 0.0896i)T \)
73 \( 1 + (-0.963 - 0.266i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.772 - 0.635i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−18.215261879127458582264895275050, −17.58090153181103709433093554026, −16.864433073184180681125802614207, −16.479575697910025864068842851825, −15.260592633312847656164618009815, −14.77869477191217610689799520173, −13.92635738915523649510609839525, −13.33139814238980521285983449438, −12.682736099283544023317144255944, −12.14886926864260096140913124527, −11.67909362854944076795278545905, −10.57912603336389570196972849365, −10.08215145213178755028442279704, −9.56003037305571157111014679161, −8.589728766707219689086722163293, −8.32106844435626076606204807011, −7.287547611764036513048518230734, −6.17562399915255024595397741687, −5.41980095967029792696118962800, −4.80622946124591057590698957542, −4.32221627402693015227009347506, −3.12917917102749721932919945686, −2.7286500486323334200374922845, −1.545596921510484570453714864018, −1.07671387037104333102549841457, 0.11755120792260424334679800843, 1.577494595618562872817617618348, 2.607205292676193757481548135156, 3.34261571554706826133317048548, 4.1571070327068399922737368182, 4.96609941403502125886314573008, 5.6478614412122023803739561983, 6.554118019684112986980579076734, 6.86001389376497127635397348129, 7.58845989352841167532117379246, 8.3949361451639052345126954962, 9.066145888040156209825271332975, 10.02168195228436439873081199386, 10.24989306001732564151565107784, 11.41464115531212453193334487998, 12.03548523767860389885762791671, 12.91624933942826358543650684703, 13.56281995577369803117339125734, 14.19281895740135466169382996074, 14.816106967262052104325239524267, 15.207506122214634330483467582228, 15.948765766721246739244845923463, 16.91702175745608352084406132275, 17.235742570255303993580576065102, 17.82036942582728871789420248646

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