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Properties

Label 1-252-252.23-r0-0-0
Degree $1$
Conductor $252$
Sign $0.235 - 0.971i$
Analytic cond. $1.17028$
Root an. cond. $1.17028$
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  + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(1.17028\)
Root analytic conductor: \(1.17028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 252,\ (0:\ ),\ 0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9307234134 - 0.7319977611i\)
\(L(\frac12)\) \(\approx\) \(0.9307234134 - 0.7319977611i\)
\(L(1)\) \(\approx\) \(1.028951323 - 0.3273427634i\)
\(L(1)\) \(\approx\) \(1.028951323 - 0.3273427634i\)

Euler product

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

−26.030324269358581804362786261703, −25.62159719841934558495116032828, −24.2780108242615751622174517053, −23.489146007133657581309039822210, −22.373581535639595205165893246734, −21.763833297919308615395704676941, −20.76814121202765894909721328065, −19.67794155954096706743127774298, −18.697539915552691333422909292576, −17.900963417614093362445644395622, −17.041053916201799700950356434735, −15.84399384743612500466103837719, −14.76696270973422192475723564567, −14.15079212672271173268007251523, −12.96141329414556695286266977866, −11.966456818460327972522050653193, −10.73410142669487925368068575230, −10.014140287882965080255544114287, −8.953497700897823036788911422782, −7.48059036239243131417451370837, −6.74984303900189984035333506065, −5.53842608403778341080908612027, −4.300796265710727327236401714256, −2.86074120806129196251579720302, −1.84132306452370481496888135302, 0.85098915330414160606247879601, 2.412805860991776896265927031375, 3.75578160107996720261186832632, 5.318601258050000278865786828728, 5.724478771508052956180639485051, 7.46536868157096858431793236224, 8.336642731265479246262764487073, 9.49391400856429514963753798535, 10.29625929567332295928616007183, 11.643820580705836612716414701933, 12.58586754791832105847465186718, 13.51829422068910302560704310275, 14.33350215469590104809536752769, 15.770257850427291422785566547489, 16.39740321920841420136676851173, 17.44739238831012860484729548006, 18.28059313247104745860592418857, 19.42395614919157392789797884267, 20.42866734648046439579311351664, 21.11742582904237845387040353856, 22.0529899905132677572609084870, 23.12096196795302518864414317192, 24.13369363185951020920645466601, 24.89171477762671425549073654166, 25.59760104537785505627538062929

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