↑

Properties

Label 1-21e2-441.173-r0-0-0
Degree $1$
Conductor $441$
Sign $0.999 + 0.0142i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.900 − 0.433i)5-s + (0.900 + 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.623 + 0.781i)11-s + (−0.365 − 0.930i)13-s + (0.0747 − 0.997i)16-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 + 0.680i)26-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.900 − 0.433i)5-s + (0.900 + 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.623 + 0.781i)11-s + (−0.365 − 0.930i)13-s + (0.0747 − 0.997i)16-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (0.955 − 0.294i)20-s + (0.955 + 0.294i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 + 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0142i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.999 + 0.0142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6183747370 + 0.004405248059i\)
\(L(\frac12)\) \(\approx\) \(0.6183747370 + 0.004405248059i\)
\(L(1)\) \(\approx\) \(0.6196179537 - 0.2008004292i\)
\(L(1)\) \(\approx\) \(0.6196179537 - 0.2008004292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (0.826 - 0.563i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.826 + 0.563i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.02470966748691760901629060322, −23.571221675921995684192866697839, −22.42831406231933035755808878259, −21.87620868442523957739672009952, −20.452645432876734178473363905579, −19.35440850014763420210995485928, −18.89624099390430577041793546010, −18.059506249919894808655741645, −16.95977146487964206019603941615, −16.21124968753947064225979683603, −15.41808261186963361025411775011, −14.72048839562164714172191621901, −13.756157725824987006088930777479, −12.836261384305658616039570427826, −11.40473231067963469396002296540, −10.793020255122152857059934293483, −9.55702168682237251904007106862, −8.5948296681069122617305428564, −7.79912397099794072487740967768, −6.9127834613666493270356182798, −6.06942408509008555702400886462, −4.77129584002030649756849214361, −3.94012042653454110479098659055, −2.44173869439744566782181144742, −0.502959890470086540029573997156, 1.026992119495278174707312594535, 2.45756019529240017004002938728, 3.49003538217013843824512521264, 4.538184910658428911068896405422, 5.38610526894785381896450822695, 7.431604211823389033787931706316, 7.78524692493176482734458377258, 8.98627667614815764608825178382, 9.81576422329721532410236914950, 10.83793718674431815008287419038, 11.644310969769910852713021787740, 12.57886878424087850891469896516, 13.05527122885672712042793630145, 14.35885635956099107482547722938, 15.520304046806806289353501571893, 16.24980509235019027200535679989, 17.44907154454194506090163957369, 18.05948001045221179900675529692, 19.065171123127624223055804980321, 19.90659669713778955096275660363, 20.446593989668102869251327809196, 21.18480763558247357462697989636, 22.54767917806436242345702677740, 22.84225329663787722719930608139, 23.91232061280672638123065026150

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