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Properties

Label 1-473-473.307-r1-0-0
Degree $1$
Conductor $473$
Sign $-0.736 + 0.675i$
Analytic cond. $50.8309$
Root an. cond. $50.8309$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.736 + 0.675i$
Analytic conductor: \(50.8309\)
Root analytic conductor: \(50.8309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (1:\ ),\ -0.736 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2616066818 - 0.6722047101i\)
\(L(\frac12)\) \(\approx\) \(-0.2616066818 - 0.6722047101i\)
\(L(1)\) \(\approx\) \(0.4257981876 - 0.3905783447i\)
\(L(1)\) \(\approx\) \(0.4257981876 - 0.3905783447i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
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 \( 1 + T \)
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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

−23.85692144465160884422781471438, −23.62849346660906155344360403846, −22.0073332806596566132117825855, −21.73988253760448933232236443684, −20.7745604116122750203854398712, −19.74640159216074315152878870645, −18.882433104415866964003879889478, −18.09640654676364861929859963755, −17.45706464164734552093713405061, −16.33753387247630803632489353943, −15.637034332868146407324419863567, −15.050372581081766116089932654381, −14.1639259677572629761234924946, −12.15257380667888996660954792026, −11.63256797145207887326634571276, −10.89253991236040474521010514376, −10.12972455347962294727610999487, −9.06794152630538219389900384576, −8.38240311352401932286037063187, −7.13058614421734165520151541124, −6.244527462251536049054034541978, −5.25674127137172546170285902344, −3.78431548707882221488747441356, −2.84658740411653765640893107602, −1.43825863504808189283627950684, 0.37219326722839659227260692944, 0.90620473456205608834243253926, 2.01680032057906862805830955022, 3.5942877266344139762105901297, 5.08805193780819795480191090219, 6.04308965598426468220746230086, 7.289616729288237635392646104319, 7.8819976136288738186359867214, 8.49290102441012553765058891268, 9.87753595444631835175169115537, 10.79175514533776973451313363276, 11.67893200118759170337015093618, 12.31383264399294263619790572412, 13.33200530662796375070057168754, 14.39477703336415988551929181392, 15.784206434684267711559568651786, 16.49412447598268557852949251474, 17.13916008881439817462951238940, 17.96472656730433086616311730745, 18.68140571145159113332149165695, 19.618059593208894769023887500545, 20.49672089798646656076659669693, 20.7671232674662437250379680083, 22.53854546750660089973316350118, 23.375100787612000049212917418900

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