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Properties

Label 2-483-161.160-c1-0-6
Degree $2$
Conductor $483$
Sign $-0.744 - 0.667i$
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  + 1.86·2-s + i·3-s + 1.46·4-s − 4.10·5-s + 1.86i·6-s + (−0.663 + 2.56i)7-s − 8-s − 9-s − 7.63·10-s + 1.89i·11-s + 1.46i·12-s + 2.18i·13-s + (−1.23 + 4.76i)14-s − 4.10i·15-s − 4.78·16-s + 1.18·17-s + ⋯
L(s)  = 1  + 1.31·2-s + 0.577i·3-s + 0.731·4-s − 1.83·5-s + 0.759i·6-s + (−0.250 + 0.968i)7-s − 0.353·8-s − 0.333·9-s − 2.41·10-s + 0.572i·11-s + 0.422i·12-s + 0.605i·13-s + (−0.329 + 1.27i)14-s − 1.05i·15-s − 1.19·16-s + 0.287·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461469 + 1.20562i\)
\(L(\frac12)\) \(\approx\) \(0.461469 + 1.20562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (0.663 - 2.56i)T \)
23 \( 1 + (4.25 + 2.20i)T \)
good2 \( 1 - 1.86T + 2T^{2} \)
5 \( 1 + 4.10T + 5T^{2} \)
11 \( 1 - 1.89iT - 11T^{2} \)
13 \( 1 - 2.18iT - 13T^{2} \)
17 \( 1 - 1.18T + 17T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
29 \( 1 - 2.39T + 29T^{2} \)
31 \( 1 - 9.32iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 5.60iT - 41T^{2} \)
43 \( 1 - 5.38iT - 43T^{2} \)
47 \( 1 - 9.57iT - 47T^{2} \)
53 \( 1 - 5.85iT - 53T^{2} \)
59 \( 1 + 3.50iT - 59T^{2} \)
61 \( 1 + 1.49T + 61T^{2} \)
67 \( 1 - 3.48iT - 67T^{2} \)
71 \( 1 - 1.81T + 71T^{2} \)
73 \( 1 + 7.97iT - 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 + 2.51T + 83T^{2} \)
89 \( 1 - 4.57T + 89T^{2} \)
97 \( 1 + 7.44T + 97T^{2} \)
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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

−11.73944788701987361249364200430, −10.82563042625524644795164787614, −9.443560735872596055695059885938, −8.649122122189060160672010346614, −7.58567131164005554894364378052, −6.51389232751841060984206424799, −5.29996695037052093839791805323, −4.51435976317239597142621569923, −3.70845310422088645837844062269, −2.83715095265614644966998497207, 0.51155809807403060200635116245, 3.15474729685199284979089818805, 3.69839733224046163664050344422, 4.66000638277408309084636787552, 5.83447230385066161551140798447, 6.96132642236813353170333803050, 7.71041554123105583409479417409, 8.416537769162600856572114891148, 9.946485754445834895710407296998, 11.24381433992092632418938242643

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