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Properties

Label 1-4729-4729.634-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.130 - 0.991i$
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  + (0.915 − 0.402i)2-s + (−0.182 + 0.983i)3-s + (0.675 − 0.737i)4-s + (−0.509 + 0.860i)5-s + (0.229 + 0.973i)6-s + (−0.639 + 0.768i)7-s + (0.321 − 0.947i)8-s + (−0.933 − 0.358i)9-s + (−0.119 + 0.992i)10-s + (0.993 − 0.111i)11-s + (0.601 + 0.798i)12-s + (0.987 + 0.158i)13-s + (−0.275 + 0.961i)14-s + (−0.753 − 0.657i)15-s + (−0.0875 − 0.996i)16-s + (0.732 + 0.681i)17-s + ⋯
L(s)  = 1  + (0.915 − 0.402i)2-s + (−0.182 + 0.983i)3-s + (0.675 − 0.737i)4-s + (−0.509 + 0.860i)5-s + (0.229 + 0.973i)6-s + (−0.639 + 0.768i)7-s + (0.321 − 0.947i)8-s + (−0.933 − 0.358i)9-s + (−0.119 + 0.992i)10-s + (0.993 − 0.111i)11-s + (0.601 + 0.798i)12-s + (0.987 + 0.158i)13-s + (−0.275 + 0.961i)14-s + (−0.753 − 0.657i)15-s + (−0.0875 − 0.996i)16-s + (0.732 + 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (634, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6888654131 - 0.7851364160i\)
\(L(\frac12)\) \(\approx\) \(0.6888654131 - 0.7851364160i\)
\(L(1)\) \(\approx\) \(1.289647990 + 0.1059497927i\)
\(L(1)\) \(\approx\) \(1.289647990 + 0.1059497927i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.915 - 0.402i)T \)
3 \( 1 + (-0.182 + 0.983i)T \)
5 \( 1 + (-0.509 + 0.860i)T \)
7 \( 1 + (-0.639 + 0.768i)T \)
11 \( 1 + (0.993 - 0.111i)T \)
13 \( 1 + (0.987 + 0.158i)T \)
17 \( 1 + (0.732 + 0.681i)T \)
19 \( 1 + (-0.305 - 0.952i)T \)
23 \( 1 + (-0.939 - 0.343i)T \)
29 \( 1 + (-0.915 - 0.402i)T \)
31 \( 1 + (-0.742 - 0.669i)T \)
37 \( 1 + (0.439 - 0.898i)T \)
41 \( 1 + (-0.821 + 0.569i)T \)
43 \( 1 + (-0.00797 - 0.999i)T \)
47 \( 1 + (0.984 + 0.174i)T \)
53 \( 1 + (-0.887 - 0.460i)T \)
59 \( 1 + (-0.996 + 0.0796i)T \)
61 \( 1 + (-0.467 + 0.884i)T \)
67 \( 1 + (-0.453 - 0.891i)T \)
71 \( 1 + (-0.944 - 0.328i)T \)
73 \( 1 + (-0.709 + 0.704i)T \)
79 \( 1 - T \)
83 \( 1 + (0.959 - 0.283i)T \)
89 \( 1 + (-0.901 - 0.431i)T \)
97 \( 1 + (-0.639 + 0.768i)T \)
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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

−18.339338144904408324346763918340, −17.39316484884209173844183828602, −16.74748174481500886182645858453, −16.46681444641408515647269063192, −15.82175821376395350870433062943, −14.84368613516783231873407713265, −14.00361744327580550614692129803, −13.73896541463846204259889086112, −12.83532810909035119945736339668, −12.53060577705690075306311004998, −11.80465467149427349976258656277, −11.32529319343672432329802181461, −10.38879784749020776250928821717, −9.24743441914813971674703148, −8.52489059469727228848236111398, −7.700730241471500493857655823261, −7.38593842967949966347629877604, −6.440821574201382156696215032060, −5.98029473311456326540282205617, −5.23734638268444028203019714836, −4.250144858200467264960325875286, −3.64185692423344050410724521957, −3.07573702332372827624460213362, −1.582632419528918668308077186605, −1.29363871471934221708342603283, 0.19292041186463252546481170522, 1.7054625348632045057503841268, 2.63910963458430007811117387517, 3.344081676038109383717440860182, 3.90563915958788475120234769823, 4.31391470980684064876184002641, 5.576039520437318273015911770727, 6.05552074175190903195428732080, 6.4829429249204594962495618010, 7.48701802385969615649514471311, 8.61931909119456078425667370386, 9.30097062435453046833563750749, 9.98170957754936062682732337139, 10.75349640983563483102330359407, 11.214894914799903882251807337095, 11.8537055474091917025253555912, 12.3671810684715835992720912520, 13.35600699046417265055922740241, 14.08090676025525907011160288346, 14.85346121506319395093387453871, 15.079484009400560660315661192768, 15.82332789056202676425664881818, 16.32584791962953537106749375744, 17.05964850493768584610516535697, 18.19905438848873549776792560908

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