Properties

Label 1-42e2-1764.1067-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.830 + 0.557i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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 + (−0.826 − 0.563i)13-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.988 − 0.149i)29-s + (−0.5 + 0.866i)31-s + (−0.988 + 0.149i)37-s + (0.955 − 0.294i)41-s + (−0.955 − 0.294i)43-s + (−0.826 − 0.563i)47-s + (0.988 + 0.149i)53-s + (−0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)5-s + (−0.900 + 0.433i)11-s + (−0.826 − 0.563i)13-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.988 − 0.149i)29-s + (−0.5 + 0.866i)31-s + (−0.988 + 0.149i)37-s + (0.955 − 0.294i)41-s + (−0.955 − 0.294i)43-s + (−0.826 − 0.563i)47-s + (0.988 + 0.149i)53-s + (−0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.830 + 0.557i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.830 + 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.221842146 + 0.3721274369i\)
\(L(\frac12)\) \(\approx\) \(1.221842146 + 0.3721274369i\)
\(L(1)\) \(\approx\) \(0.8606524421 + 0.1288154784i\)
\(L(1)\) \(\approx\) \(0.8606524421 + 0.1288154784i\)

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 \( 1 + (-0.826 - 0.563i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (0.988 - 0.149i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (0.955 - 0.294i)T \)
43 \( 1 + (-0.955 - 0.294i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (-0.955 - 0.294i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.826 + 0.563i)T \)
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

−19.790670147908920949558204354869, −19.440545322416027105975394697318, −18.64671012007065295797375071972, −17.55661164280577489728960606156, −17.07170377270086744261640256629, −16.32326582848209305708891242247, −15.60653878064726416568761131244, −14.92526913424302128318436854329, −13.96269350612978041398643797677, −13.07447707589999318195037150260, −12.70091783770405733899174899587, −11.76036571849582212073406483056, −11.05639273277659261082706106043, −10.11626140164680385361884215059, −9.317543826169680697807227114485, −8.558024681584854388622491061347, −7.885634699304576407529394237044, −7.0538804824366288310719983264, −5.98344421043302211138405992464, −5.136369706185165152421894335164, −4.554895366673536252997950346160, −3.556271207816605112058828260504, −2.50985948432030319619715290812, −1.51975058590595943049236612626, −0.44424424928609989393163871310, 0.47680417218371376668502521672, 1.999035649541277744255753808001, 2.78765144071095878027728864118, 3.44664340113632431221726108985, 4.68463421538496549601446984872, 5.321606871536558547171195627964, 6.41390246548691944692732821226, 7.17961469870022023884065702913, 7.763363036600371635692484731041, 8.621431380584186440853451527387, 9.817962874257532519079601183526, 10.35756969180440316490397116825, 10.904158800220534959214412122363, 12.07532471585778834449701330705, 12.447579463165114764077556874736, 13.53192112409818227842555060786, 14.30727671016785349284111625188, 14.93804469224769295911579797044, 15.56653427279778842923536299707, 16.389552142024500342426192875303, 17.25493225075161682055362068744, 18.19540159105853129618562140357, 18.43915932811567412093687211488, 19.40682664129487838623136871426, 20.00747223028210894107697371368

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