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Properties

Label 1-4851-4851.1984-r1-0-0
Degree $1$
Conductor $4851$
Sign $0.845 - 0.533i$
Analytic cond. $521.312$
Root an. cond. $521.312$
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  + (−0.925 + 0.379i)2-s + (0.712 − 0.701i)4-s + (0.995 + 0.0896i)5-s + (−0.393 + 0.919i)8-s + (−0.955 + 0.294i)10-s + (0.925 − 0.379i)13-s + (0.0149 − 0.999i)16-s + (0.447 − 0.894i)17-s + (−0.913 + 0.406i)19-s + (0.772 − 0.635i)20-s + (0.623 + 0.781i)23-s + (0.983 + 0.178i)25-s + (−0.712 + 0.701i)26-s + (−0.842 − 0.538i)29-s + (0.978 + 0.207i)31-s + (0.365 + 0.930i)32-s + ⋯
L(s)  = 1  + (−0.925 + 0.379i)2-s + (0.712 − 0.701i)4-s + (0.995 + 0.0896i)5-s + (−0.393 + 0.919i)8-s + (−0.955 + 0.294i)10-s + (0.925 − 0.379i)13-s + (0.0149 − 0.999i)16-s + (0.447 − 0.894i)17-s + (−0.913 + 0.406i)19-s + (0.772 − 0.635i)20-s + (0.623 + 0.781i)23-s + (0.983 + 0.178i)25-s + (−0.712 + 0.701i)26-s + (−0.842 − 0.538i)29-s + (0.978 + 0.207i)31-s + (0.365 + 0.930i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(521.312\)
Root analytic conductor: \(521.312\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (1984, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (1:\ ),\ 0.845 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.793151853 - 0.5180700494i\)
\(L(\frac12)\) \(\approx\) \(1.793151853 - 0.5180700494i\)
\(L(1)\) \(\approx\) \(0.9075118119 + 0.05952383857i\)
\(L(1)\) \(\approx\) \(0.9075118119 + 0.05952383857i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.925 + 0.379i)T \)
5 \( 1 + (0.995 + 0.0896i)T \)
13 \( 1 + (0.925 - 0.379i)T \)
17 \( 1 + (0.447 - 0.894i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.842 - 0.538i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.842 - 0.538i)T \)
41 \( 1 + (-0.992 + 0.119i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (-0.525 + 0.850i)T \)
53 \( 1 + (0.998 + 0.0598i)T \)
59 \( 1 + (-0.992 - 0.119i)T \)
61 \( 1 + (0.447 - 0.894i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.936 - 0.351i)T \)
73 \( 1 + (0.999 - 0.0299i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.925 + 0.379i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−18.130323933687235402097594012242, −17.27317083766165031615407430894, −16.82188805014245986503366157454, −16.418606227163613108049300766531, −15.290729170219126744190574432, −14.89556331517911428312991353252, −13.78989931358724079053863693408, −13.21020825046351758793797939711, −12.62469677498366745664854507429, −11.84998714497469036201188947939, −10.989983314721691961654909186547, −10.48783904055136714093004319877, −9.945472399941928158837228139267, −9.07451211546890063291855348157, −8.6032841392334225429069357334, −8.05293526905563108065635741428, −6.739416454324775113271355761150, −6.60366202497978937055154644630, −5.70607074764717230709614236030, −4.73757464604960633844455526861, −3.74805796100279332164970461336, −3.017067633296961901516847347552, −2.07178095805264558295092765965, −1.56177936293877726746146826490, −0.73330254251971664405666739783, 0.43766787731287856660249616637, 1.318534395385638655817042603871, 1.96047644075499190287272871375, 2.8288026888931033328871280486, 3.665782691116968612782660506590, 5.06896251614327435198923330675, 5.46951059531566826624176688992, 6.34510642520226936247060038670, 6.75385920300762416408419038898, 7.70377202109567081609331748881, 8.358502594360185020377621875038, 9.11454294288557673595754657850, 9.614567389957023321707857560636, 10.37079379416616194426873577542, 10.843094591987259300984599173926, 11.62869068549614983700971288372, 12.449857483186017518346674910089, 13.495109449714911675798489535705, 13.782558975364908277077124857322, 14.74500604900632713698701576472, 15.27035832447742524859582759668, 16.018931165941947926194372583682, 16.71487653230274816483726490257, 17.29323547302408504321997904423, 17.774963680329552479567590657811

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