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Properties

Label 1-4729-4729.156-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.559 + 0.828i$
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 + (−0.635 − 0.772i)5-s + (−0.336 − 0.941i)6-s + (−0.0186 − 0.999i)7-s + (0.908 + 0.417i)8-s + (−0.563 + 0.826i)9-s + (−0.518 − 0.855i)10-s + (0.158 − 0.987i)11-s + (−0.198 − 0.980i)12-s + (−0.929 + 0.368i)13-s + (0.124 − 0.992i)14-s + (−0.385 + 0.922i)15-s + (0.839 + 0.543i)16-s + (0.995 − 0.0981i)17-s + ⋯
L(s)  = 1  + (0.989 + 0.143i)2-s + (−0.467 − 0.884i)3-s + (0.959 + 0.283i)4-s + (−0.635 − 0.772i)5-s + (−0.336 − 0.941i)6-s + (−0.0186 − 0.999i)7-s + (0.908 + 0.417i)8-s + (−0.563 + 0.826i)9-s + (−0.518 − 0.855i)10-s + (0.158 − 0.987i)11-s + (−0.198 − 0.980i)12-s + (−0.929 + 0.368i)13-s + (0.124 − 0.992i)14-s + (−0.385 + 0.922i)15-s + (0.839 + 0.543i)16-s + (0.995 − 0.0981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.559 + 0.828i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.559 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3236538654 - 0.6090737932i\)
\(L(\frac12)\) \(\approx\) \(-0.3236538654 - 0.6090737932i\)
\(L(1)\) \(\approx\) \(1.064223478 - 0.5897897358i\)
\(L(1)\) \(\approx\) \(1.064223478 - 0.5897897358i\)

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 \( 1 + (-0.635 - 0.772i)T \)
7 \( 1 + (-0.0186 - 0.999i)T \)
11 \( 1 + (0.158 - 0.987i)T \)
13 \( 1 + (-0.929 + 0.368i)T \)
17 \( 1 + (0.995 - 0.0981i)T \)
19 \( 1 + (-0.988 + 0.153i)T \)
23 \( 1 + (0.748 - 0.663i)T \)
29 \( 1 + (-0.143 - 0.989i)T \)
31 \( 1 + (-0.930 + 0.366i)T \)
37 \( 1 + (-0.853 - 0.520i)T \)
41 \( 1 + (0.0318 + 0.999i)T \)
43 \( 1 + (-0.926 - 0.375i)T \)
47 \( 1 + (-0.811 + 0.584i)T \)
53 \( 1 + (0.0849 + 0.996i)T \)
59 \( 1 + (-0.756 - 0.653i)T \)
61 \( 1 + (-0.838 - 0.545i)T \)
67 \( 1 + (-0.295 - 0.955i)T \)
71 \( 1 + (0.842 + 0.538i)T \)
73 \( 1 + (0.997 + 0.0690i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.706 - 0.708i)T \)
89 \( 1 + (-0.993 + 0.114i)T \)
97 \( 1 + (0.0186 + 0.999i)T \)
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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

−18.67167346812610249562701079213, −17.910084733637547960104323636310, −17.01988558614265088256447139646, −16.465712440452507621220975268917, −15.524452156867752286449486736490, −15.15148512102121961077050812527, −14.79435720918258531257053821833, −14.31218089261587153973854726414, −12.98753451870325386849461329566, −12.3212039538406016847930988658, −11.96696471826177340346635054896, −11.28780187600832818398642973263, −10.56342548871671220792669479940, −10.03610362076370071510214535321, −9.29852781939227992383554571942, −8.26669629787407273993247968433, −7.27470214893734296935298101144, −6.79371121760867994983450533353, −5.91446423952243771729183773890, −5.16508133180141129897303383490, −4.796156122899785609639570665021, −3.778724213514820902251450770581, −3.27132977993598690114543680854, −2.54544540739910334399742121265, −1.65364205188662177812603727772, 0.1321790012340895641735745486, 1.13185001095404481455251159618, 1.86182007818605110240743540949, 2.98920772355979181715957912661, 3.691492826803526062354964641065, 4.57440972950806581773405476386, 5.05632672128219212562184420080, 5.89856833183796297041139418443, 6.64210736747293108206649946308, 7.246863996405304841058882017644, 7.93002063359653284668688389662, 8.37663249293465141340835658450, 9.57599987319968328323545815863, 10.80003987728028725047799677498, 11.01538421779357890588785319974, 11.95141165759608312312938506277, 12.368982806337662596655731073226, 12.980904230354824098043021174780, 13.58134393962240055361515877876, 14.2626737188131264906587526943, 14.791676261920140153088596851390, 15.8397016581726732391164062497, 16.63575635889102595343106608317, 16.85748159891608231727759187206, 17.21939448665093302956301680215

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