Properties

Label 2-483-1.1-c3-0-57
Degree $2$
Conductor $483$
Sign $-1$
Analytic cond. $28.4979$
Root an. cond. $5.33834$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.55·2-s + 3·3-s − 5.57·4-s + 9.75·5-s + 4.66·6-s − 7·7-s − 21.1·8-s + 9·9-s + 15.1·10-s − 48.6·11-s − 16.7·12-s + 21.0·13-s − 10.8·14-s + 29.2·15-s + 11.7·16-s − 48.9·17-s + 14.0·18-s − 35.9·19-s − 54.4·20-s − 21·21-s − 75.7·22-s − 23·23-s − 63.3·24-s − 29.8·25-s + 32.6·26-s + 27·27-s + 39.0·28-s + ⋯
L(s)  = 1  + 0.550·2-s + 0.577·3-s − 0.697·4-s + 0.872·5-s + 0.317·6-s − 0.377·7-s − 0.933·8-s + 0.333·9-s + 0.479·10-s − 1.33·11-s − 0.402·12-s + 0.448·13-s − 0.207·14-s + 0.503·15-s + 0.183·16-s − 0.698·17-s + 0.183·18-s − 0.434·19-s − 0.608·20-s − 0.218·21-s − 0.733·22-s − 0.208·23-s − 0.539·24-s − 0.238·25-s + 0.246·26-s + 0.192·27-s + 0.263·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(28.4979\)
Root analytic conductor: \(5.33834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 483,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
7 \( 1 + 7T \)
23 \( 1 + 23T \)
good2 \( 1 - 1.55T + 8T^{2} \)
5 \( 1 - 9.75T + 125T^{2} \)
11 \( 1 + 48.6T + 1.33e3T^{2} \)
13 \( 1 - 21.0T + 2.19e3T^{2} \)
17 \( 1 + 48.9T + 4.91e3T^{2} \)
19 \( 1 + 35.9T + 6.85e3T^{2} \)
29 \( 1 + 157.T + 2.43e4T^{2} \)
31 \( 1 + 141.T + 2.97e4T^{2} \)
37 \( 1 + 58.2T + 5.06e4T^{2} \)
41 \( 1 + 253.T + 6.89e4T^{2} \)
43 \( 1 + 89.4T + 7.95e4T^{2} \)
47 \( 1 - 471.T + 1.03e5T^{2} \)
53 \( 1 - 461.T + 1.48e5T^{2} \)
59 \( 1 + 775.T + 2.05e5T^{2} \)
61 \( 1 + 304.T + 2.26e5T^{2} \)
67 \( 1 + 192.T + 3.00e5T^{2} \)
71 \( 1 + 19.8T + 3.57e5T^{2} \)
73 \( 1 + 1.10e3T + 3.89e5T^{2} \)
79 \( 1 - 757.T + 4.93e5T^{2} \)
83 \( 1 - 148.T + 5.71e5T^{2} \)
89 \( 1 + 1.40e3T + 7.04e5T^{2} \)
97 \( 1 + 41.6T + 9.12e5T^{2} \)
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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

−10.03629838532778585714031700240, −9.184462122179408590608198851514, −8.533132065291115411508675987693, −7.40372806838696290055718392540, −6.08512196606651966832478885690, −5.38676096228515099189684961170, −4.25429012729173676755101026487, −3.15470112844774025343378443854, −2.02968768820791289761346491683, 0, 2.02968768820791289761346491683, 3.15470112844774025343378443854, 4.25429012729173676755101026487, 5.38676096228515099189684961170, 6.08512196606651966832478885690, 7.40372806838696290055718392540, 8.533132065291115411508675987693, 9.184462122179408590608198851514, 10.03629838532778585714031700240

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