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Properties

Label 2-483-69.68-c1-0-3
Degree $2$
Conductor $483$
Sign $0.999 + 0.0209i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.26i·2-s + (−1.42 − 0.984i)3-s − 3.12·4-s − 3.16·5-s + (−2.22 + 3.22i)6-s + i·7-s + 2.54i·8-s + (1.05 + 2.80i)9-s + 7.15i·10-s + 5.18·11-s + (4.45 + 3.07i)12-s − 6.74·13-s + 2.26·14-s + (4.50 + 3.11i)15-s − 0.486·16-s + 2.12·17-s + ⋯
L(s)  = 1  − 1.60i·2-s + (−0.822 − 0.568i)3-s − 1.56·4-s − 1.41·5-s + (−0.910 + 1.31i)6-s + 0.377i·7-s + 0.899i·8-s + (0.353 + 0.935i)9-s + 2.26i·10-s + 1.56·11-s + (1.28 + 0.888i)12-s − 1.87·13-s + 0.605·14-s + (1.16 + 0.804i)15-s − 0.121·16-s + 0.515·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.999 + 0.0209i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.999 + 0.0209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257737 - 0.00269736i\)
\(L(\frac12)\) \(\approx\) \(0.257737 - 0.00269736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.42 + 0.984i)T \)
7 \( 1 - iT \)
23 \( 1 + (2.64 + 4.00i)T \)
good2 \( 1 + 2.26iT - 2T^{2} \)
5 \( 1 + 3.16T + 5T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
13 \( 1 + 6.74T + 13T^{2} \)
17 \( 1 - 2.12T + 17T^{2} \)
19 \( 1 - 3.62iT - 19T^{2} \)
29 \( 1 - 0.746iT - 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 5.13iT - 37T^{2} \)
41 \( 1 - 3.15iT - 41T^{2} \)
43 \( 1 - 8.15iT - 43T^{2} \)
47 \( 1 - 1.22iT - 47T^{2} \)
53 \( 1 + 9.28T + 53T^{2} \)
59 \( 1 - 5.21iT - 59T^{2} \)
61 \( 1 - 4.15iT - 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 8.44iT - 71T^{2} \)
73 \( 1 - 1.29T + 73T^{2} \)
79 \( 1 + 4.40iT - 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 - 1.68T + 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−11.45343533026969886706893555353, −10.30285090255644402002877990381, −9.611503518845790243719352233034, −8.347363148421479965090646766011, −7.42337419060420053096975881251, −6.37806736353050764706931986620, −4.77900637668502745282608291010, −4.10568490450529394402404213953, −2.78989053258806985098921993897, −1.32615454970279055949747265870, 0.19340348168219587240280419108, 3.76085569129296590939768079482, 4.49877575695148770109167921278, 5.33081211906254352215875358928, 6.59584020967741694535940405117, 7.18004793393161715187928113465, 7.892455030203698513472666386841, 9.117636294035967579663006654171, 9.783959478387219408483958009163, 11.14656970199983622523963530356

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