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Properties

Label 1-1176-1176.1163-r0-0-0
Degree $1$
Conductor $1176$
Sign $0.967 - 0.253i$
Analytic cond. $5.46132$
Root an. cond. $5.46132$
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 + (0.900 − 0.433i)13-s + (−0.623 − 0.781i)17-s + 19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.623 + 0.781i)29-s − 31-s + (−0.623 − 0.781i)37-s + (0.222 − 0.974i)41-s + (−0.222 − 0.974i)43-s + (−0.900 + 0.433i)47-s + (0.623 − 0.781i)53-s + (0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)5-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (−0.623 − 0.781i)17-s + 19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.623 + 0.781i)29-s − 31-s + (−0.623 − 0.781i)37-s + (0.222 − 0.974i)41-s + (−0.222 − 0.974i)43-s + (−0.900 + 0.433i)47-s + (0.623 − 0.781i)53-s + (0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(5.46132\)
Root analytic conductor: \(5.46132\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (1163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (0:\ ),\ 0.967 - 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.558118386 - 0.2008971193i\)
\(L(\frac12)\) \(\approx\) \(1.558118386 - 0.2008971193i\)
\(L(1)\) \(\approx\) \(1.135781521 + 0.01966916840i\)
\(L(1)\) \(\approx\) \(1.135781521 + 0.01966916840i\)

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 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.623 - 0.781i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
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 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + T \)
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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

−21.283482761957818229783310389858, −20.35680644972131320342437570317, −19.85615012171287382004496547234, −19.13037875585614454174458559606, −18.089618470892667264739760302991, −17.33478569613195452345791694205, −16.67118521566113825284602389734, −15.8814779866156938601360305465, −15.21375354591739134328806389402, −14.22171757726819322837495828793, −13.33951643494222543839879859753, −12.78826035519076621445159867448, −11.6683919948064993137743803194, −11.40596916027738471146673053157, −10.055220310640859966595411061020, −9.24470809281101238105755324102, −8.65526044382222275252179024561, −7.77603745447842775325546170387, −6.75694404404620591033375092141, −5.92016884675427364311838854631, −4.90041698357562235909683211436, −4.128645297922814462147226812115, −3.328502869445219920596656460586, −1.76186687676593796099006227719, −1.13231714754479596370343122801, 0.763436207843219868719814993461, 2.087113286050543442104112432794, 3.23393738358318548464044059607, 3.701606012565017308501588803376, 4.971252408193467851992612424024, 5.97616400110600247056186172060, 6.81283899299287791243878295733, 7.3893664364938131814795835460, 8.584858071762284623922178753236, 9.19352784269855678740768528254, 10.33722961165287266741166812878, 11.01561983761608574773759529306, 11.611180189520913375156015039988, 12.546122897563292234813130788822, 13.66096081706641537544673814033, 14.16025558334513536656980400747, 14.9560091244003281291368302346, 15.85173735380626483639201557806, 16.38911269016780142383813907437, 17.57174315869004552692083612962, 18.18445250639471081044667360719, 18.8017344447642266244207856693, 19.687541284915293838896161105314, 20.33049738489465915423870616002, 21.2476880342044447491088668699

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