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Properties

Label 1-4729-4729.1016-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.954 + 0.296i$
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  + (0.783 − 0.620i)2-s + (−0.984 − 0.174i)3-s + (0.229 − 0.973i)4-s + (0.103 + 0.994i)5-s + (−0.880 + 0.474i)6-s + (−0.0875 − 0.996i)7-s + (−0.424 − 0.905i)8-s + (0.939 + 0.343i)9-s + (0.698 + 0.715i)10-s + (−0.954 − 0.298i)11-s + (−0.395 + 0.918i)12-s + (−0.999 − 0.0159i)13-s + (−0.687 − 0.726i)14-s + (0.0717 − 0.997i)15-s + (−0.894 − 0.446i)16-s + (−0.971 + 0.236i)17-s + ⋯
L(s)  = 1  + (0.783 − 0.620i)2-s + (−0.984 − 0.174i)3-s + (0.229 − 0.973i)4-s + (0.103 + 0.994i)5-s + (−0.880 + 0.474i)6-s + (−0.0875 − 0.996i)7-s + (−0.424 − 0.905i)8-s + (0.939 + 0.343i)9-s + (0.698 + 0.715i)10-s + (−0.954 − 0.298i)11-s + (−0.395 + 0.918i)12-s + (−0.999 − 0.0159i)13-s + (−0.687 − 0.726i)14-s + (0.0717 − 0.997i)15-s + (−0.894 − 0.446i)16-s + (−0.971 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.954 + 0.296i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.954 + 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5350090177 + 0.08126252755i\)
\(L(\frac12)\) \(\approx\) \(0.5350090177 + 0.08126252755i\)
\(L(1)\) \(\approx\) \(0.7700310675 - 0.3917058448i\)
\(L(1)\) \(\approx\) \(0.7700310675 - 0.3917058448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.783 - 0.620i)T \)
3 \( 1 + (-0.984 - 0.174i)T \)
5 \( 1 + (0.103 + 0.994i)T \)
7 \( 1 + (-0.0875 - 0.996i)T \)
11 \( 1 + (-0.954 - 0.298i)T \)
13 \( 1 + (-0.999 - 0.0159i)T \)
17 \( 1 + (-0.971 + 0.236i)T \)
19 \( 1 + (-0.481 - 0.876i)T \)
23 \( 1 + (-0.275 - 0.961i)T \)
29 \( 1 + (0.783 + 0.620i)T \)
31 \( 1 + (0.763 + 0.645i)T \)
37 \( 1 + (-0.993 + 0.111i)T \)
41 \( 1 + (-0.366 - 0.930i)T \)
43 \( 1 + (-0.453 + 0.891i)T \)
47 \( 1 + (0.601 - 0.798i)T \)
53 \( 1 + (-0.998 - 0.0478i)T \)
59 \( 1 + (0.00797 + 0.999i)T \)
61 \( 1 + (0.410 + 0.912i)T \)
67 \( 1 + (0.495 + 0.868i)T \)
71 \( 1 + (-0.614 + 0.788i)T \)
73 \( 1 + (0.522 - 0.852i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.336 - 0.941i)T \)
89 \( 1 + (0.351 - 0.936i)T \)
97 \( 1 + (-0.0875 - 0.996i)T \)
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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

−17.71358300076797388657166369933, −17.37091651738273397705667045434, −16.76039832507858278948581751325, −15.9005834064933274692675892573, −15.57506281143394028917730172593, −15.15975496808567435160122249531, −14.021356004809803046309513835923, −13.26401871667574880585302804260, −12.658394097058095273602444871016, −12.174426966475067993023330897032, −11.76787395043308389851474438901, −10.874094102486260051975612977924, −9.81468222970576308472597279295, −9.356884969221596634519554391481, −8.242790252525095624800856564790, −7.87651309018751891334308832776, −6.81080826521682903999146031873, −6.152864895991112394777430433684, −5.49687922305443195447002972252, −4.91743273483532665061909362299, −4.57646560019607000552371278086, −3.581033004191041695275139648434, −2.40122593117131106540869856167, −1.83818100645157744234476062940, −0.16912782865877962658519902146, 0.67375700249872661764893702809, 1.892523657171014582114474199080, 2.568480690172925031868334119, 3.32164226779633316839965988459, 4.41070432550057932220884106024, 4.73543274503175510859209991722, 5.59989302757661265865926131276, 6.53956909424101301353339317603, 6.82835223031501044746437657522, 7.45344358005105333957255009, 8.66533749646125087093626588047, 9.972878693612333704785201009642, 10.40083478921329457157692278272, 10.671069981145141230208408573611, 11.37475637209800017929605476079, 12.092479785151933600090247010196, 12.83571772505117393923513538306, 13.41450012547912792349331930928, 13.97401110231741010684628529744, 14.73246710412646302343658931795, 15.536119633593683674517682628689, 15.979074373885270423665114089001, 16.95559445548159995600343955336, 17.71022043834365971134254555819, 18.08970272176807753946596137567

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