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Properties

Label 1-4729-4729.2599-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.999 + 0.0267i$
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  + (0.00797 + 0.999i)2-s + (0.803 + 0.595i)3-s + (−0.999 + 0.0159i)4-s + (0.721 + 0.692i)5-s + (−0.589 + 0.808i)6-s + (0.949 − 0.313i)7-s + (−0.0239 − 0.999i)8-s + (0.290 + 0.956i)9-s + (−0.687 + 0.726i)10-s + (0.182 + 0.983i)11-s + (−0.812 − 0.582i)12-s + (0.467 − 0.884i)13-s + (0.321 + 0.947i)14-s + (0.166 + 0.986i)15-s + (0.999 − 0.0318i)16-s + (0.848 + 0.529i)17-s + ⋯
L(s)  = 1  + (0.00797 + 0.999i)2-s + (0.803 + 0.595i)3-s + (−0.999 + 0.0159i)4-s + (0.721 + 0.692i)5-s + (−0.589 + 0.808i)6-s + (0.949 − 0.313i)7-s + (−0.0239 − 0.999i)8-s + (0.290 + 0.956i)9-s + (−0.687 + 0.726i)10-s + (0.182 + 0.983i)11-s + (−0.812 − 0.582i)12-s + (0.467 − 0.884i)13-s + (0.321 + 0.947i)14-s + (0.166 + 0.986i)15-s + (0.999 − 0.0318i)16-s + (0.848 + 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.999 + 0.0267i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (2599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.999 + 0.0267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03916927803 + 2.928307805i\)
\(L(\frac12)\) \(\approx\) \(-0.03916927803 + 2.928307805i\)
\(L(1)\) \(\approx\) \(0.9821994358 + 1.296410471i\)
\(L(1)\) \(\approx\) \(0.9821994358 + 1.296410471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.00797 + 0.999i)T \)
3 \( 1 + (0.803 + 0.595i)T \)
5 \( 1 + (0.721 + 0.692i)T \)
7 \( 1 + (0.949 - 0.313i)T \)
11 \( 1 + (0.182 + 0.983i)T \)
13 \( 1 + (0.467 - 0.884i)T \)
17 \( 1 + (0.848 + 0.529i)T \)
19 \( 1 + (-0.921 + 0.388i)T \)
23 \( 1 + (-0.991 - 0.127i)T \)
29 \( 1 + (-0.00797 + 0.999i)T \)
31 \( 1 + (0.830 - 0.556i)T \)
37 \( 1 + (0.260 + 0.965i)T \)
41 \( 1 + (-0.939 - 0.343i)T \)
43 \( 1 + (-0.839 + 0.543i)T \)
47 \( 1 + (-0.997 - 0.0637i)T \)
53 \( 1 + (0.993 - 0.111i)T \)
59 \( 1 + (0.856 + 0.516i)T \)
61 \( 1 + (0.894 + 0.446i)T \)
67 \( 1 + (-0.773 - 0.633i)T \)
71 \( 1 + (0.522 - 0.852i)T \)
73 \( 1 + (0.959 + 0.283i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.244 - 0.969i)T \)
89 \( 1 + (-0.742 - 0.669i)T \)
97 \( 1 + (0.949 - 0.313i)T \)
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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

−18.09642907009581205855062683773, −17.36764971480856650667487437722, −16.74069402061126396523262790901, −15.803718342915035167803423215178, −14.70247310329815581670870319672, −14.18286107526510869985925359634, −13.7060264042259161546887526132, −13.24541469651086746764077401865, −12.37810827159804995196972647764, −11.76004917960495742361761071276, −11.33063107181473073799663176117, −10.19391700245501123135213665842, −9.632733630449398221380844091393, −8.82094972315951987831413064024, −8.43659977799810496934342889362, −8.00091701217277469534769717657, −6.700576016981018194108370478461, −5.888223189073207532791803414856, −5.17402188156986229114277417043, −4.27173353996629155353628259418, −3.673920210936159431441132186145, −2.60322912666545859485151517543, −2.05620028518537292075508815670, −1.40966064274041242906867632803, −0.69162044672380769316869083516, 1.37200881966002358386820059723, 2.02889554529205701496822326737, 3.16345696374452513012257335387, 3.84499103488719002488931808016, 4.60323914388186185054042743342, 5.27034296666392531407874270958, 6.03169658954333300431152990560, 6.851765340229145257817955620939, 7.60809262100679601650715167973, 8.24161774590879028594364843692, 8.6325636140618292576556531199, 9.812358300854329764258533127871, 10.14400919368826136459061219743, 10.56706254518613653326683024109, 11.75711905673801160468542737082, 12.846122943195398407726326830500, 13.38082161368338745544494890207, 14.11626497625614486378820414762, 14.664470391514905885043709963288, 14.98502722391181980061868377161, 15.518098116041043743583377360233, 16.636655509519726591055780792809, 16.99481617286434635148326816292, 17.8629519535960677083843021177, 18.227935677394853450815802129909

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