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Properties

Label 1-476-476.67-r1-0-0
Degree $1$
Conductor $476$
Sign $-0.832 + 0.553i$
Analytic cond. $51.1533$
Root an. cond. $51.1533$
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)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s − 15-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s − 15-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(51.1533\)
Root analytic conductor: \(51.1533\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 476,\ (1:\ ),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2478570087 - 0.8201367013i\)
\(L(\frac12)\) \(\approx\) \(-0.2478570087 - 0.8201367013i\)
\(L(1)\) \(\approx\) \(0.7193573933 - 0.4785035105i\)
\(L(1)\) \(\approx\) \(0.7193573933 - 0.4785035105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
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 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−23.8159803953123037573669016298, −23.02633170661374243779950385688, −22.429850392903795122248847428148, −21.67693938714164284311132039989, −20.72224133541281184675385967160, −20.25125091590810270594490385124, −18.56063158383593089219154190588, −18.23981432101096415069151448544, −17.24226472427788976238155334800, −16.34471719014834902746442177667, −15.4697757471890309482903944137, −14.73322250038827617888443663564, −13.891641341987043452350942851350, −12.7245822276267158142136226000, −11.66293222393781655668554741962, −10.74205897230193830184340103051, −10.15404239204692421203323573598, −9.36035722879784066006152641132, −8.11258589396778725535191987174, −6.844747493288581332908067955298, −6.0078309028052806353972186230, −5.12053086471068188161307322937, −3.9283269944952226482762003938, −3.01394538549633208597060743873, −1.637710569187547066400598594441, 0.24361814737000040360682326601, 1.23430478323269516465396987555, 2.29298220708738374148146252485, 3.75845966517495909372375281555, 5.32698241816411253382046016580, 5.680499822749489990369834947246, 6.809996345624059474148684795716, 7.96921242561921823433061086670, 8.68079477449359084325472101303, 9.77258348779047158603758613322, 11.08897761012925206258730180072, 11.61475045013754826029734716705, 12.89141663230206188762693150002, 13.33393233711064257800377840148, 14.003781355242867143038815416342, 15.57903243917055866060551730747, 16.40654121779344300973727146509, 17.08747576860251310494353055094, 18.07101777600551626321164452981, 18.58755415391494705715295005796, 19.712919577706015381802022108837, 20.47584978176176692278663684749, 21.5016142504854923137025708521, 22.198609279306759438258905246530, 23.43717143021714189625351893435

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