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Properties

Label 1-4729-4729.1004-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.752 + 0.658i$
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  + (−0.0239 − 0.999i)2-s + (−0.336 + 0.941i)3-s + (−0.998 + 0.0478i)4-s + (−0.663 + 0.748i)5-s + (0.949 + 0.313i)6-s + (0.576 − 0.817i)7-s + (0.0717 + 0.997i)8-s + (−0.773 − 0.633i)9-s + (0.763 + 0.645i)10-s + (−0.522 − 0.852i)11-s + (0.290 − 0.956i)12-s + (−0.993 + 0.111i)13-s + (−0.830 − 0.556i)14-s + (−0.481 − 0.876i)15-s + (0.995 − 0.0955i)16-s + (−0.103 + 0.994i)17-s + ⋯
L(s)  = 1  + (−0.0239 − 0.999i)2-s + (−0.336 + 0.941i)3-s + (−0.998 + 0.0478i)4-s + (−0.663 + 0.748i)5-s + (0.949 + 0.313i)6-s + (0.576 − 0.817i)7-s + (0.0717 + 0.997i)8-s + (−0.773 − 0.633i)9-s + (0.763 + 0.645i)10-s + (−0.522 − 0.852i)11-s + (0.290 − 0.956i)12-s + (−0.993 + 0.111i)13-s + (−0.830 − 0.556i)14-s + (−0.481 − 0.876i)15-s + (0.995 − 0.0955i)16-s + (−0.103 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.752 + 0.658i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.752 + 0.658i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3704930568 + 0.1392871315i\)
\(L(\frac12)\) \(\approx\) \(0.3704930568 + 0.1392871315i\)
\(L(1)\) \(\approx\) \(0.5764463971 - 0.1151400181i\)
\(L(1)\) \(\approx\) \(0.5764463971 - 0.1151400181i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.0239 - 0.999i)T \)
3 \( 1 + (-0.336 + 0.941i)T \)
5 \( 1 + (-0.663 + 0.748i)T \)
7 \( 1 + (0.576 - 0.817i)T \)
11 \( 1 + (-0.522 - 0.852i)T \)
13 \( 1 + (-0.993 + 0.111i)T \)
17 \( 1 + (-0.103 + 0.994i)T \)
19 \( 1 + (-0.366 + 0.930i)T \)
23 \( 1 + (-0.927 - 0.373i)T \)
29 \( 1 + (0.0239 - 0.999i)T \)
31 \( 1 + (-0.198 - 0.980i)T \)
37 \( 1 + (-0.709 - 0.704i)T \)
41 \( 1 + (-0.495 - 0.868i)T \)
43 \( 1 + (0.150 + 0.988i)T \)
47 \( 1 + (-0.981 - 0.190i)T \)
53 \( 1 + (0.944 - 0.328i)T \)
59 \( 1 + (-0.0557 + 0.998i)T \)
61 \( 1 + (0.182 + 0.983i)T \)
67 \( 1 + (0.467 - 0.884i)T \)
71 \( 1 + (-0.996 - 0.0796i)T \)
73 \( 1 + (0.651 + 0.758i)T \)
79 \( 1 - T \)
83 \( 1 + (0.675 + 0.737i)T \)
89 \( 1 + (0.589 - 0.808i)T \)
97 \( 1 + (0.576 - 0.817i)T \)
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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

−17.866281020610168857109089496812, −17.54608625386441358013291793765, −16.81264147984584456702253959871, −15.98096716221831528302217450026, −15.559830252650731146943208355714, −14.78029095945022133820892111252, −14.19032217610617933624341531256, −13.2739095061116749945066672506, −12.76682996173496506009529002033, −12.05138515495616450279207839184, −11.774288901008224360912438234145, −10.636552269755939396799788539522, −9.63004657863194135583904142159, −8.88909983247817715735775065323, −8.32069912150026360221214845582, −7.6595223532273038108634391859, −7.160098500348043674937023763748, −6.5150840357528954973786334109, −5.289895252482484420219428860027, −5.09912453964639761471463675321, −4.605343643858113815414377625986, −3.23987183909061514117410929344, −2.23736289453848217165021040435, −1.38694784269691186704410141285, −0.212574647498267131912251449776, 0.52786911103590852602534637050, 1.93071921773913734061092073894, 2.69043479546032592606425284845, 3.70360568201475639071229786817, 3.95620773557330187852442109792, 4.62391332985328282641296050738, 5.56786092828311189470337211400, 6.23869979925110304804274259904, 7.491718316765662026673513617670, 8.12798078317173850501713694228, 8.65489187918545282620150166045, 9.85360718447840403272168869042, 10.307506895853969292999939817208, 10.65511754330351203421484626545, 11.43746215352848252745711585641, 11.81449239917453868653458960268, 12.63941325687449778734728929187, 13.6010762256549448807569311412, 14.358900741447005068225171294771, 14.694692519320266404511531706287, 15.42537647008364231304846271712, 16.45123853450285901186141865992, 16.898800727362021573478255117935, 17.63386868941702743397764138892, 18.260557328908374882902937448930

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