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Properties

Label 2-483-69.68-c1-0-43
Degree $2$
Conductor $483$
Sign $-0.372 - 0.927i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.372 - 0.927i$
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: \((2,\ 483,\ (\ :1/2),\ -0.372 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402085 + 0.594802i\)
\(L(\frac12)\) \(\approx\) \(0.402085 + 0.594802i\)
\(L(\frac{3}{2})\) 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

−10.43588844620891900303383178381, −10.02053857630984249174147654532, −9.056471205794827497506682136477, −7.55588612333207953065503629384, −6.59283453930268221922704529356, −5.20677171827493069119803322087, −4.75948648792440304115126292212, −2.57717815444779222827688624975, −2.17560480631001152089451434450, −0.44409404792207332614813165245, 2.49596522087233862277351810724, 4.72701973397698985493582654707, 5.31852790786017196997752710730, 5.85131283366733195380480884535, 6.82891139811884960089434654908, 7.72267222670285758435083216937, 8.912663448835078720138331972178, 9.883698445164001948189038932552, 10.17226081811149273711801484426, 11.55697413156535480595007966111

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