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Properties

Label 1-42e2-1764.443-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.239 - 0.970i$
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  + (0.222 − 0.974i)5-s + (−0.900 + 0.433i)11-s + (0.0747 − 0.997i)13-s + (0.988 − 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.365 + 0.930i)29-s + (0.5 + 0.866i)31-s + (0.365 − 0.930i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (0.0747 − 0.997i)47-s + (−0.365 − 0.930i)53-s + (0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)5-s + (−0.900 + 0.433i)11-s + (0.0747 − 0.997i)13-s + (0.988 − 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.365 + 0.930i)29-s + (0.5 + 0.866i)31-s + (0.365 − 0.930i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (0.0747 − 0.997i)47-s + (−0.365 − 0.930i)53-s + (0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.239 - 0.970i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.239 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.222130497 - 0.9569378292i\)
\(L(\frac12)\) \(\approx\) \(1.222130497 - 0.9569378292i\)
\(L(1)\) \(\approx\) \(1.071142434 - 0.2833923509i\)
\(L(1)\) \(\approx\) \(1.071142434 - 0.2833923509i\)

Euler product

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

−20.53796114233431988127355858472, −19.37363979396469477959669005328, −18.87946803957119627249996005097, −18.39680433474889739025354866742, −17.41934399499038597287004361248, −16.852674052732044881820552140649, −15.747423234426412666773092329923, −15.35225331716170051134218265547, −14.33841062860484062572791542429, −13.78242909850345393863844516591, −13.14151310717882660404459644446, −12.05029185382080904873165224890, −11.20265881511590960126247210524, −10.82822751608390959254281983747, −9.67428804764176811757413868943, −9.34502160202123135531859537643, −7.92553396165703452870208279327, −7.54141644098206246182130403027, −6.504320514468323934937409262461, −5.86295201572081129441934267823, −4.94597978039550572322476760731, −3.87830023494252521396030914332, −2.96102189315740445433194246311, −2.34553717740128385348987546098, −1.085263723545538055564262368459, 0.636947779749049283372874733157, 1.57871732815541696494158975815, 2.703649877652221494080891611564, 3.57644598418051072651707986721, 4.73689115667254299882738436129, 5.33542189073609156195565126048, 5.95154201403386536987424392938, 7.27011608300994417861510854481, 7.92352423834882745632723179782, 8.62054391859300183953544602976, 9.546715871396978040695223755510, 10.24295327283714146135131639740, 10.912563125666321314201368015912, 12.26479885662608114144728099863, 12.48349802387375452528604782310, 13.24287851430570759163107889376, 14.1250014700981215719749125029, 14.91273225920930471327076794706, 15.804518046479628230728412116970, 16.39602236542001416039058760813, 17.04007142770381840442913166332, 18.034479180110803453908096922680, 18.36241497922511343400159681561, 19.51077001864255170269429275541, 20.18287294284671990089385994750

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