Properties

Label 2-483-1.1-c1-0-20
Degree $2$
Conductor $483$
Sign $1$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 4·5-s + 2·6-s − 7-s + 9-s + 8·10-s − 5·11-s + 2·12-s − 2·13-s − 2·14-s + 4·15-s − 4·16-s + 2·18-s − 5·19-s + 8·20-s − 21-s − 10·22-s − 23-s + 11·25-s − 4·26-s + 27-s − 2·28-s − 2·29-s + 8·30-s + 6·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 1.78·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 2.52·10-s − 1.50·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 1.03·15-s − 16-s + 0.471·18-s − 1.14·19-s + 1.78·20-s − 0.218·21-s − 2.13·22-s − 0.208·23-s + 11/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 1.46·30-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{483} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.767442407\)
\(L(\frac12)\) \(\approx\) \(3.767442407\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.94122298294594700260446625869, −10.06348584847165124277699778695, −9.421863300952081605429638012938, −8.292339792365995708433110594415, −6.91642082849532995462484101470, −6.00450627059691435421291295880, −5.33569718385275672521817298149, −4.33710151569730855874203864959, −2.74710044114612301763782652477, −2.32541152601280590701225190627, 2.32541152601280590701225190627, 2.74710044114612301763782652477, 4.33710151569730855874203864959, 5.33569718385275672521817298149, 6.00450627059691435421291295880, 6.91642082849532995462484101470, 8.292339792365995708433110594415, 9.421863300952081605429638012938, 10.06348584847165124277699778695, 10.94122298294594700260446625869

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