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Properties

Label 1-4729-4729.1014-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.230 - 0.973i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.944 + 0.328i)2-s + (−0.0875 + 0.996i)3-s + (0.783 − 0.620i)4-s + (0.208 − 0.978i)5-s + (−0.244 − 0.969i)6-s + (−0.976 − 0.216i)7-s + (−0.536 + 0.843i)8-s + (−0.984 − 0.174i)9-s + (0.124 + 0.992i)10-s + (0.150 − 0.988i)11-s + (0.549 + 0.835i)12-s + (−0.869 + 0.493i)13-s + (0.993 − 0.116i)14-s + (0.956 + 0.293i)15-s + (0.229 − 0.973i)16-s + (0.800 − 0.599i)17-s + ⋯
L(s)  = 1  + (−0.944 + 0.328i)2-s + (−0.0875 + 0.996i)3-s + (0.783 − 0.620i)4-s + (0.208 − 0.978i)5-s + (−0.244 − 0.969i)6-s + (−0.976 − 0.216i)7-s + (−0.536 + 0.843i)8-s + (−0.984 − 0.174i)9-s + (0.124 + 0.992i)10-s + (0.150 − 0.988i)11-s + (0.549 + 0.835i)12-s + (−0.869 + 0.493i)13-s + (0.993 − 0.116i)14-s + (0.956 + 0.293i)15-s + (0.229 − 0.973i)16-s + (0.800 − 0.599i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.230 - 0.973i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4726303378 - 0.3736378879i\)
\(L(\frac12)\) \(\approx\) \(0.4726303378 - 0.3736378879i\)
\(L(1)\) \(\approx\) \(0.5872415018 + 0.04808784832i\)
\(L(1)\) \(\approx\) \(0.5872415018 + 0.04808784832i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.944 + 0.328i)T \)
3 \( 1 + (-0.0875 + 0.996i)T \)
5 \( 1 + (0.208 - 0.978i)T \)
7 \( 1 + (-0.976 - 0.216i)T \)
11 \( 1 + (0.150 - 0.988i)T \)
13 \( 1 + (-0.869 + 0.493i)T \)
17 \( 1 + (0.800 - 0.599i)T \)
19 \( 1 + (0.999 + 0.0106i)T \)
23 \( 1 + (-0.601 + 0.798i)T \)
29 \( 1 + (0.944 + 0.328i)T \)
31 \( 1 + (-0.939 - 0.343i)T \)
37 \( 1 + (-0.836 + 0.547i)T \)
41 \( 1 + (0.563 - 0.826i)T \)
43 \( 1 + (0.999 - 0.0265i)T \)
47 \( 1 + (-0.833 - 0.551i)T \)
53 \( 1 + (0.853 + 0.520i)T \)
59 \( 1 + (0.964 - 0.262i)T \)
61 \( 1 + (0.890 - 0.455i)T \)
67 \( 1 + (0.867 - 0.497i)T \)
71 \( 1 + (-0.997 - 0.0690i)T \)
73 \( 1 + (-0.859 - 0.511i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.419 + 0.907i)T \)
89 \( 1 + (0.904 + 0.427i)T \)
97 \( 1 + (-0.976 - 0.216i)T \)
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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

−18.25528215892866016347595778399, −17.719594898301026941558699672111, −17.41183747684748315270220715051, −16.38185428729146708319343084712, −15.88695400641266466317125223691, −14.735412904582923648547928631181, −14.52909524634364077213323251866, −13.3851761554642167058236955431, −12.60707624291278011413222363308, −12.27007712463221768683540960988, −11.63881456940936391252625155964, −10.705293727225166388669857432474, −9.98140168990048250192016574210, −9.73605252951626002911948231430, −8.70896998985875351107201853727, −7.85943699364949466109956523134, −7.246247368447918446380837046, −6.876203538661512981486109748303, −6.10342745921716895396661048476, −5.45335877228468469022570943265, −3.906450515529789493932891959723, −3.0346438938701898998548783688, −2.556159328344899578367414361035, −1.895204570883463488051977769883, −0.86947881300842658822252594715, 0.305157323634132618709933917701, 1.06054027964699528178991044914, 2.29220487771835099097647848925, 3.20209186214462192481479695577, 3.86165377777302783574329940782, 5.05981454806943370006331002350, 5.48604487821973095718114074759, 6.128756158655116166243870234019, 7.0907291491217856894292216997, 7.84234129687442422865887496214, 8.72418927346161786176500928305, 9.225064504286128134903878980, 9.76674942077559554613066767870, 10.14901201448222533069302939677, 11.09986242198750448322897883218, 11.9085035208221137741112335290, 12.25049934669721081001340033533, 13.63031904625146672600206389143, 14.02313641157741468438122633085, 14.84087541886521117818291908691, 15.87597276683799342044520817749, 16.20891463665739825906692853631, 16.42005769759622033877914930261, 17.2319792580230166036141385764, 17.702283555598554576588825687521

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