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Properties

Label 1-4729-4729.242-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: $\chi_{4729} (242, \cdot )$
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

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

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