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Properties

Label 1-4729-4729.1058-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.618 + 0.785i$
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  + (−0.971 + 0.236i)2-s + (0.959 + 0.283i)3-s + (0.887 − 0.460i)4-s + (−0.563 − 0.826i)5-s + (−0.998 − 0.0478i)6-s + (−0.989 + 0.143i)7-s + (−0.753 + 0.657i)8-s + (0.839 + 0.543i)9-s + (0.742 + 0.669i)10-s + (−0.709 − 0.704i)11-s + (0.981 − 0.190i)12-s + (0.439 − 0.898i)13-s + (0.927 − 0.373i)14-s + (−0.305 − 0.952i)15-s + (0.576 − 0.817i)16-s + (−0.509 − 0.860i)17-s + ⋯
L(s)  = 1  + (−0.971 + 0.236i)2-s + (0.959 + 0.283i)3-s + (0.887 − 0.460i)4-s + (−0.563 − 0.826i)5-s + (−0.998 − 0.0478i)6-s + (−0.989 + 0.143i)7-s + (−0.753 + 0.657i)8-s + (0.839 + 0.543i)9-s + (0.742 + 0.669i)10-s + (−0.709 − 0.704i)11-s + (0.981 − 0.190i)12-s + (0.439 − 0.898i)13-s + (0.927 − 0.373i)14-s + (−0.305 − 0.952i)15-s + (0.576 − 0.817i)16-s + (−0.509 − 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.618 + 0.785i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.618 + 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7094180754 + 0.3443852258i\)
\(L(\frac12)\) \(\approx\) \(0.7094180754 + 0.3443852258i\)
\(L(1)\) \(\approx\) \(0.7103270112 + 0.02244252320i\)
\(L(1)\) \(\approx\) \(0.7103270112 + 0.02244252320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.971 + 0.236i)T \)
3 \( 1 + (0.959 + 0.283i)T \)
5 \( 1 + (-0.563 - 0.826i)T \)
7 \( 1 + (-0.989 + 0.143i)T \)
11 \( 1 + (-0.709 - 0.704i)T \)
13 \( 1 + (0.439 - 0.898i)T \)
17 \( 1 + (-0.509 - 0.860i)T \)
19 \( 1 + (0.821 + 0.569i)T \)
23 \( 1 + (-0.773 - 0.633i)T \)
29 \( 1 + (-0.971 - 0.236i)T \)
31 \( 1 + (0.410 + 0.912i)T \)
37 \( 1 + (0.0398 + 0.999i)T \)
41 \( 1 + (-0.453 - 0.891i)T \)
43 \( 1 + (-0.0557 + 0.998i)T \)
47 \( 1 + (-0.336 + 0.941i)T \)
53 \( 1 + (-0.978 + 0.205i)T \)
59 \( 1 + (-0.848 - 0.529i)T \)
61 \( 1 + (0.260 + 0.965i)T \)
67 \( 1 + (-0.150 + 0.988i)T \)
71 \( 1 + (0.698 + 0.715i)T \)
73 \( 1 + (-0.687 + 0.726i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.424 + 0.905i)T \)
89 \( 1 + (-0.999 - 0.0159i)T \)
97 \( 1 + (-0.989 + 0.143i)T \)
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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

−18.38182026955502489102453245968, −17.68306403646732042120673132153, −16.69561313231829522164978138333, −15.93893387416817446889956912608, −15.406099126693437119989865085594, −15.06753632139990939690342469587, −13.976044494150897753376959751990, −13.32745485148563376440306434436, −12.641913953524522943154704699803, −11.92564556065303424361654440450, −11.165996201195334625176706391504, −10.439311109761385963054448853157, −9.719450933373501477181076628843, −9.335678446975783183449792227, −8.47145955962040035644664091886, −7.69182903420803952676318200973, −7.29189063272409849828171072392, −6.6340439467754221433319907231, −6.02213515315696114059534233492, −4.36938822159377049500285278283, −3.56230255760840551937977300975, −3.21560927302084409658414355661, −2.17500328490261852847653417676, −1.82918200452982381414513339619, −0.345619447340448508472307732565, 0.71235584629096879241546763347, 1.61077363387916815381790811123, 2.887071771279851525478092148177, 3.04711492215466092694529719886, 4.07457276635678872704733574499, 5.15777016777510361408077697035, 5.774675980350428935664716017710, 6.75390854888201174595694391517, 7.58019766802522068935311251714, 8.17840797730616629833417599629, 8.55063917677858681305040858655, 9.38254925743897712656539123678, 9.819391828077189396297252192002, 10.55460338181551902311599506003, 11.31972692457772630420929449482, 12.231325808110771199603835861640, 12.89663074296883867193444764370, 13.577299978742093349323648831127, 14.29791214241208899960658889453, 15.43360223163599397712677633047, 15.637237296271455049190557443350, 16.23292997740691751227474672825, 16.54729928226249735907865790624, 17.68187908472971847183307426893, 18.51007958236840490321397275851

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