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Properties

Label 1-4729-4729.104-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.917 + 0.397i$
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  + (0.989 − 0.143i)2-s + (−0.467 + 0.884i)3-s + (0.959 − 0.283i)4-s + (0.986 + 0.164i)5-s + (−0.336 + 0.941i)6-s + (0.875 − 0.483i)7-s + (0.908 − 0.417i)8-s + (−0.563 − 0.826i)9-s + (0.999 + 0.0212i)10-s + (−0.158 − 0.987i)11-s + (−0.198 + 0.980i)12-s + (0.145 + 0.989i)13-s + (0.796 − 0.604i)14-s + (−0.606 + 0.795i)15-s + (0.839 − 0.543i)16-s + (0.412 + 0.910i)17-s + ⋯
L(s)  = 1  + (0.989 − 0.143i)2-s + (−0.467 + 0.884i)3-s + (0.959 − 0.283i)4-s + (0.986 + 0.164i)5-s + (−0.336 + 0.941i)6-s + (0.875 − 0.483i)7-s + (0.908 − 0.417i)8-s + (−0.563 − 0.826i)9-s + (0.999 + 0.0212i)10-s + (−0.158 − 0.987i)11-s + (−0.198 + 0.980i)12-s + (0.145 + 0.989i)13-s + (0.796 − 0.604i)14-s + (−0.606 + 0.795i)15-s + (0.839 − 0.543i)16-s + (0.412 + 0.910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.917 + 0.397i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.917 + 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.515106062 + 0.9360115431i\)
\(L(\frac12)\) \(\approx\) \(4.515106062 + 0.9360115431i\)
\(L(1)\) \(\approx\) \(2.314767183 + 0.3032841209i\)
\(L(1)\) \(\approx\) \(2.314767183 + 0.3032841209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.989 - 0.143i)T \)
3 \( 1 + (-0.467 + 0.884i)T \)
5 \( 1 + (0.986 + 0.164i)T \)
7 \( 1 + (0.875 - 0.483i)T \)
11 \( 1 + (-0.158 - 0.987i)T \)
13 \( 1 + (0.145 + 0.989i)T \)
17 \( 1 + (0.412 + 0.910i)T \)
19 \( 1 + (0.361 + 0.932i)T \)
23 \( 1 + (-0.748 - 0.663i)T \)
29 \( 1 + (0.143 - 0.989i)T \)
31 \( 1 + (0.930 + 0.366i)T \)
37 \( 1 + (0.877 + 0.479i)T \)
41 \( 1 + (-0.0318 + 0.999i)T \)
43 \( 1 + (-0.137 + 0.990i)T \)
47 \( 1 + (0.100 - 0.994i)T \)
53 \( 1 + (-0.820 - 0.571i)T \)
59 \( 1 + (-0.187 - 0.982i)T \)
61 \( 1 + (0.0531 + 0.998i)T \)
67 \( 1 + (-0.679 - 0.733i)T \)
71 \( 1 + (0.0451 + 0.998i)T \)
73 \( 1 + (-0.558 - 0.829i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.966 + 0.257i)T \)
89 \( 1 + (-0.397 - 0.917i)T \)
97 \( 1 + (-0.875 + 0.483i)T \)
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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

−17.92575171020306770064144853976, −17.52315249036834407776597061575, −16.97141597929236927358923926316, −15.91455223473781379360938223660, −15.44674646855339862136289306492, −14.47710844812934892554423065339, −14.02695631596193301696221026323, −13.40040534839595445532982245544, −12.7558831421323702335616934256, −12.208942579677637161732496685914, −11.62909148629091687806536728464, −10.86177488445974758010449310475, −10.20676453024966852609888807021, −9.21632313023678647541526693189, −8.25198702566178016371316158369, −7.48662272645337310192604114379, −7.10436381014948705993119260052, −6.05922678548688911861342513070, −5.59439489660487199933159929409, −5.04260031894743545651607663033, −4.48497633675788218568863471259, −2.9550049638874037262421513281, −2.49426630871785707539100642334, −1.73171656090632111855620874331, −1.03611402699688208842146042477, 1.08906887143046814075011904887, 1.752511924483215325040060984494, 2.77678739464560755177049336659, 3.55009125254083866133287096890, 4.35430058324117053102697011388, 4.77808119759946880395629087699, 5.825082549100885253054753852503, 6.03014235797599177365041026048, 6.731212842714560169763276237286, 7.98072201219609110856135797359, 8.551254608760463669959738943227, 9.92170626817844428263653177865, 10.01815845003268479090410726484, 10.879227964795275881359412430323, 11.436690271766940935836315859682, 11.946153559766042896719884040409, 12.92698438937486330285986341953, 13.72724765871697411538912667272, 14.25999617605411468743541778660, 14.57813296624642375912076909492, 15.39582493011740527994795651998, 16.43047426931681213667436373510, 16.57967462330571097041727991924, 17.263566244861450753933536645403, 18.14501657197179218458750281750

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