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Properties

Label 1-4729-4729.1029-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.245 - 0.969i$
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  + (−0.978 − 0.205i)2-s + (−0.639 + 0.768i)3-s + (0.915 + 0.402i)4-s + (0.495 − 0.868i)5-s + (0.783 − 0.620i)6-s + (−0.424 + 0.905i)7-s + (−0.812 − 0.582i)8-s + (−0.182 − 0.983i)9-s + (−0.663 + 0.748i)10-s + (0.0557 − 0.998i)11-s + (−0.894 + 0.446i)12-s + (−0.996 + 0.0796i)13-s + (0.601 − 0.798i)14-s + (0.351 + 0.936i)15-s + (0.675 + 0.737i)16-s + (0.366 + 0.930i)17-s + ⋯
L(s)  = 1  + (−0.978 − 0.205i)2-s + (−0.639 + 0.768i)3-s + (0.915 + 0.402i)4-s + (0.495 − 0.868i)5-s + (0.783 − 0.620i)6-s + (−0.424 + 0.905i)7-s + (−0.812 − 0.582i)8-s + (−0.182 − 0.983i)9-s + (−0.663 + 0.748i)10-s + (0.0557 − 0.998i)11-s + (−0.894 + 0.446i)12-s + (−0.996 + 0.0796i)13-s + (0.601 − 0.798i)14-s + (0.351 + 0.936i)15-s + (0.675 + 0.737i)16-s + (0.366 + 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.245 - 0.969i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5282068359 - 0.4110436180i\)
\(L(\frac12)\) \(\approx\) \(0.5282068359 - 0.4110436180i\)
\(L(1)\) \(\approx\) \(0.5738159223 - 0.03826197082i\)
\(L(1)\) \(\approx\) \(0.5738159223 - 0.03826197082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.978 - 0.205i)T \)
3 \( 1 + (-0.639 + 0.768i)T \)
5 \( 1 + (0.495 - 0.868i)T \)
7 \( 1 + (-0.424 + 0.905i)T \)
11 \( 1 + (0.0557 - 0.998i)T \)
13 \( 1 + (-0.996 + 0.0796i)T \)
17 \( 1 + (0.366 + 0.930i)T \)
19 \( 1 + (-0.589 - 0.808i)T \)
23 \( 1 + (0.984 - 0.174i)T \)
29 \( 1 + (0.978 - 0.205i)T \)
31 \( 1 + (0.933 - 0.358i)T \)
37 \( 1 + (-0.848 - 0.529i)T \)
41 \( 1 + (0.954 + 0.298i)T \)
43 \( 1 + (0.709 - 0.704i)T \)
47 \( 1 + (0.0875 + 0.996i)T \)
53 \( 1 + (0.971 - 0.236i)T \)
59 \( 1 + (0.0398 - 0.999i)T \)
61 \( 1 + (-0.856 - 0.516i)T \)
67 \( 1 + (0.522 + 0.852i)T \)
71 \( 1 + (0.166 + 0.986i)T \)
73 \( 1 + (0.380 - 0.924i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.989 - 0.143i)T \)
89 \( 1 + (-0.975 + 0.221i)T \)
97 \( 1 + (-0.424 + 0.905i)T \)
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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

−18.13708707317310887102678161331, −17.696438203797325547305554336723, −16.959073744506433237647977102357, −16.82234374879714195441348736724, −15.76229403772109641684870765301, −15.03312707275714578742173067016, −14.21051661232301904125434466446, −13.80132363303140416091902335940, −12.6838290860957801993286584079, −12.19176878746460468880064658838, −11.41279762047663172281379381598, −10.62929807523271751843102467994, −10.106824440616072015700726499599, −9.782177893600352551896866538689, −8.69464341443685403230277676842, −7.60996738673163252628004582635, −7.24434338733264383577934756502, −6.84082202999359002145244048872, −6.16357071073686208343490818648, −5.32416910021792974991019238122, −4.490152267545074659818675274123, −3.02678783413997813074352933349, −2.514396467209945641353512081133, −1.62143617332569768567459838245, −0.84395459673511618327488481327, 0.37234941148424591307613955472, 1.137764551631285209848512051458, 2.365926258295714247498971090914, 2.90351190578005304713865319473, 3.952714985567943449520537953526, 4.854903688973738723758176423408, 5.64002408817113945169522499618, 6.171500083591511749100592801212, 6.8269972629270466134764417304, 8.06664636087881233764927965413, 8.80593908845764662922521495931, 9.06525131815225309834375727681, 9.84311344157359575316123561973, 10.38205664547926883245770510622, 11.16571088191215722726823662522, 11.81424408692826142565458210042, 12.56539284782915115599652134565, 12.81432675626742938563401500269, 14.12269242646557984383764479443, 15.01106970207514285518511832452, 15.68553833333026864492938379205, 16.08792814656971130284804656560, 16.9215199866489185957127248768, 17.24190363832247296418961758662, 17.711332928408117633638478606664

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