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Properties

Label 1-21e2-441.349-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.0782 - 0.996i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
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  + (−0.733 + 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.988 − 0.149i)5-s + (0.623 + 0.781i)8-s + (−0.623 + 0.781i)10-s + (−0.733 + 0.680i)11-s + (−0.955 − 0.294i)13-s + (−0.988 − 0.149i)16-s + (0.900 + 0.433i)17-s − 19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (0.900 − 0.433i)26-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.988 − 0.149i)5-s + (0.623 + 0.781i)8-s + (−0.623 + 0.781i)10-s + (−0.733 + 0.680i)11-s + (−0.955 − 0.294i)13-s + (−0.988 − 0.149i)16-s + (0.900 + 0.433i)17-s − 19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (0.900 − 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0782 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0782 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0782 - 0.996i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.0782 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4220838247 - 0.4565261460i\)
\(L(\frac12)\) \(\approx\) \(0.4220838247 - 0.4565261460i\)
\(L(1)\) \(\approx\) \(0.7199074684 + 0.1019144813i\)
\(L(1)\) \(\approx\) \(0.7199074684 + 0.1019144813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.733 + 0.680i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.955 - 0.294i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (0.0747 + 0.997i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (0.988 - 0.149i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (-0.365 - 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.955 + 0.294i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−24.26274730883905478251215298932, −23.03755209275483153877380948488, −22.00655021247232706867773114480, −21.211000100350942103202567902384, −20.95371423436154739497416533338, −19.537316023577353996294898032920, −18.97839020790368115252369555041, −18.07181557079067616828813997485, −17.25921199458836104315713740200, −16.65274910667124418840746120273, −15.54287099816877805065054324287, −14.21276361157528847973377277742, −13.42990678120568463902576631430, −12.531852514455421375044676913923, −11.57384691267138813731379872371, −10.51249505864845787297186115543, −9.91769614623376692108391951716, −9.05147446264497951191641091205, −8.01517909007208766009737738716, −7.04402805044962736752973609383, −5.84658599465001816496386314914, −4.68663111121762853995326654854, −3.16895631088071562836755062152, −2.3966945126254769310416971794, −1.24823283358778154504859898499, 0.208246729519571783407884952527, 1.68948919215829712013373164828, 2.59709178962254463563027187953, 4.6454209176423847627705452425, 5.430128644191826361760173760062, 6.38071491691664955917860568460, 7.3550703359228197856628895605, 8.311987886048891911715520521014, 9.29115731788801734497720465940, 10.21745829365807469572373883654, 10.59526014829716867123757231156, 12.30983387554915129754231474205, 13.10572412255329623682970693248, 14.35982356200510284166653527781, 14.83028234604028659304236136577, 15.9180423848202952214928050079, 17.005399242655559371185503657368, 17.33538477376100549513005438518, 18.34248906549725323014694163125, 19.03638084114684165274763031186, 20.16966883489312715284458182707, 20.89775280286692747339996204625, 21.938525727852650441766502034579, 22.96938352717467777134264664421, 23.80646571887648591027107377726

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