Properties

Label 1-252-252.227-r1-0-0
Degree $1$
Conductor $252$
Sign $-0.458 + 0.888i$
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  + (−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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.458 + 0.888i$
Analytic conductor: \(27.0811\)
Root analytic conductor: \(27.0811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 252,\ (1:\ ),\ -0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2484478681 + 0.4075206943i\)
\(L(\frac12)\) \(\approx\) \(0.2484478681 + 0.4075206943i\)
\(L(1)\) \(\approx\) \(0.7907034266 - 0.03781058501i\)
\(L(1)\) \(\approx\) \(0.7907034266 - 0.03781058501i\)

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

−25.87505773217464799117902871503, −24.30705071808903533900624913267, −23.70160682374584608275684235764, −22.793912248357329479705600350284, −21.705316685934601426282161118979, −21.15564741682697162593537694495, −19.57451070146478598870594766707, −19.164166201209901754844013129495, −18.13929549531195033645475749197, −17.16545075392343735713844445175, −15.86229437960432378520807958950, −15.34159520306584521142761188769, −14.069698516924633252426659533758, −13.38580875714753798116911669724, −11.94303176598986172514225018469, −11.10664543861043454366244513065, −10.347190709027036637386486871448, −8.90655364188514677856921073268, −7.97954770557799207887553889220, −6.79278164887131216835646457803, −5.945142777472883732438476287996, −4.348531599335571830444611368905, −3.34728069849768380191016065454, −2.078568792008952522269063120991, −0.15591989955179243436356448824, 1.2973809522454353692044145328, 2.86564603234880974206475568883, 4.31676739604396183493955386505, 5.11049723001625909465004162929, 6.4703793498766130039048479589, 7.818412974242298233275190696190, 8.49287272280258096259417276368, 9.74126623568093585094636656762, 10.737788132536120400453885828, 12.03520689411020001081330229619, 12.68819507151495308017419205569, 13.6711048615128556147193154249, 15.015727492879536005667433869486, 15.80284613640714156745184781966, 16.63826873484480193582576266721, 17.76665562146938067870346823840, 18.57518332635985411566792185738, 19.914316697754158948915890919457, 20.42450816477675436400557270721, 21.26249871841605852003621282149, 22.749117085366976714740208437654, 23.1699871152372641375766592005, 24.34183406456582302972488646997, 25.056935459775185846612440046635, 26.001162804959397480640875179606

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