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Properties

Label 1-252-252.47-r1-0-0
Degree $1$
Conductor $252$
Sign $0.954 - 0.296i$
Analytic cond. $27.0811$
Root an. cond. $27.0811$
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  + 5-s + 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + 23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯
L(s)  = 1  + 5-s + 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + 23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.954 - 0.296i$
Analytic conductor: \(27.0811\)
Root analytic conductor: \(27.0811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 252,\ (1:\ ),\ 0.954 - 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.508523831 - 0.3807728517i\)
\(L(\frac12)\) \(\approx\) \(2.508523831 - 0.3807728517i\)
\(L(1)\) \(\approx\) \(1.424630130 - 0.08374978030i\)
\(L(1)\) \(\approx\) \(1.424630130 - 0.08374978030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−25.5731229405380519006278783027, −25.092001753956209662582984094552, −24.12878528515679725870832232230, −22.94773670674630996414182235595, −22.13322635123082805316056901962, −21.19405107842663446205928383568, −20.50529276616881905001140070390, −19.22249224468283852683807407944, −18.4324271566895965919424434287, −17.3056646850617682662357894177, −16.72939646322463162048894185406, −15.523208996629740850808560313708, −14.22497995706335652027547591941, −13.79482446404834224589580386060, −12.565377817688203145784808901874, −11.513278557746209410942054828510, −10.43509533847752078737402603563, −9.32571076264836077145156564817, −8.6900435051785087346579008946, −6.98235158340065028647598113176, −6.2767851915760928034923639745, −5.04945463071927421221085892932, −3.80594172891834726031570525757, −2.31019947325711441483748942643, −1.182771545181253320972486563, 0.98610389728980673000786825142, 2.248813092421947544685447261655, 3.611632300735445413511609638595, 4.997717761326149312149352775547, 6.10924300695193401513161308552, 6.92809744932843406801132729559, 8.52080638013237273054745966254, 9.25506257069568273785904956558, 10.43212903920015501707332956428, 11.24151969461444932078731383756, 12.72622888173333428891262326840, 13.31345530706049482043208552772, 14.47915901495050699065151522625, 15.25508870143546358473230643898, 16.61084901683925725074996933517, 17.41491004411398754862958685981, 18.094613608732053967114993063834, 19.353306694888743768031567418614, 20.19387482414888384187587049873, 21.28050584697390598160451350382, 21.98693022430332205815530005598, 22.85414705242200593177756072112, 24.03298731405237975023681832413, 24.9725080650587184833644192526, 25.59552396101861538439352534632

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