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Properties

Label 1-42e2-1764.1363-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.960 - 0.277i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (−0.955 − 0.294i)5-s + (−0.0747 − 0.997i)11-s + (−0.0747 − 0.997i)13-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.365 + 0.930i)29-s + 31-s + (−0.988 − 0.149i)37-s + (0.733 − 0.680i)41-s + (0.733 + 0.680i)43-s + (−0.900 − 0.433i)47-s + (−0.988 + 0.149i)53-s + (−0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)5-s + (−0.0747 − 0.997i)11-s + (−0.0747 − 0.997i)13-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.365 + 0.930i)29-s + 31-s + (−0.988 − 0.149i)37-s + (0.733 − 0.680i)41-s + (0.733 + 0.680i)43-s + (−0.900 − 0.433i)47-s + (−0.988 + 0.149i)53-s + (−0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.960 - 0.277i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.960 - 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09410314754 - 0.6644263211i\)
\(L(\frac12)\) \(\approx\) \(0.09410314754 - 0.6644263211i\)
\(L(1)\) \(\approx\) \(0.7510365973 - 0.2513979193i\)
\(L(1)\) \(\approx\) \(0.7510365973 - 0.2513979193i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.955 - 0.294i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (0.733 + 0.680i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 - T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (-0.826 - 0.563i)T \)
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

−20.70835701216903622420630256369, −19.54493863320250850126581073839, −19.25767543679430729121441932501, −18.56583833365588814158535447413, −17.49162064409942557825463203729, −17.01282419753932438845062514896, −15.998723002444272239963411599331, −15.38831542271609066047617380732, −14.74304517563125960770090952946, −14.06712572139246976864879044913, −12.94722047973026510923870254127, −12.32343753076939528136586562125, −11.63697859860986989293281247682, −10.85105070176850062410201298907, −10.10360429675886089113162208296, −9.20106965209162365126186337059, −8.299758834506713243351193625620, −7.649816413147445987210654128418, −6.77260414233220923342308645562, −6.18993752213167220427387839351, −4.75434057869649677421184997982, −4.30616878785495244049531492181, −3.42335327885697938630815337756, −2.34581113578707875813024847455, −1.41236259247039749761840787274, 0.27010856096100319035594025361, 1.148952103851435135694491767732, 2.840705417344010288423095758046, 3.16664834402208692692280697308, 4.4335234200254868402615836223, 5.00269502494412071070376461168, 5.98835339649072337169220152694, 7.031716734278582351104090100366, 7.63745189887925075338484024204, 8.71421138499674879990385892238, 8.88814684398888170055006181768, 10.27115815434792026235731749211, 11.00410131292287441013060418258, 11.5448154263568993615218227547, 12.4896254566453106845719570864, 13.102052949188393669766807566767, 13.92240292283008828090245896445, 14.84506674733038049842777456830, 15.635932032475428344611006898199, 16.02794914133590411086732619816, 16.91274375933095863623840267356, 17.70388029860570816273831626883, 18.49378373743268970534811206369, 19.43696347113914283338853153941, 19.64509997807191965388869753940

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