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Properties

Label 2-483-161.9-c1-0-8
Degree $2$
Conductor $483$
Sign $-0.560 - 0.827i$
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  + (−0.187 + 0.772i)2-s + (−0.327 + 0.945i)3-s + (1.21 + 0.626i)4-s + (2.15 + 0.860i)5-s + (−0.669 − 0.429i)6-s + (0.216 + 2.63i)7-s + (−1.75 + 2.02i)8-s + (−0.786 − 0.618i)9-s + (−1.06 + 1.50i)10-s + (0.0463 + 0.191i)11-s + (−0.989 + 0.943i)12-s + (−0.365 − 0.799i)13-s + (−2.07 − 0.327i)14-s + (−1.51 + 1.75i)15-s + (0.350 + 0.492i)16-s + (0.361 − 7.59i)17-s + ⋯
L(s)  = 1  + (−0.132 + 0.546i)2-s + (−0.188 + 0.545i)3-s + (0.607 + 0.313i)4-s + (0.961 + 0.385i)5-s + (−0.273 − 0.175i)6-s + (0.0818 + 0.996i)7-s + (−0.620 + 0.715i)8-s + (−0.262 − 0.206i)9-s + (−0.337 + 0.474i)10-s + (0.0139 + 0.0576i)11-s + (−0.285 + 0.272i)12-s + (−0.101 − 0.221i)13-s + (−0.555 − 0.0874i)14-s + (−0.391 + 0.451i)15-s + (0.0877 + 0.123i)16-s + (0.0877 − 1.84i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.560 - 0.827i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.560 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771729 + 1.45488i\)
\(L(\frac12)\) \(\approx\) \(0.771729 + 1.45488i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 - 0.945i)T \)
7 \( 1 + (-0.216 - 2.63i)T \)
23 \( 1 + (-4.26 + 2.18i)T \)
good2 \( 1 + (0.187 - 0.772i)T + (-1.77 - 0.916i)T^{2} \)
5 \( 1 + (-2.15 - 0.860i)T + (3.61 + 3.45i)T^{2} \)
11 \( 1 + (-0.0463 - 0.191i)T + (-9.77 + 5.04i)T^{2} \)
13 \( 1 + (0.365 + 0.799i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-0.361 + 7.59i)T + (-16.9 - 1.61i)T^{2} \)
19 \( 1 + (-0.172 - 3.62i)T + (-18.9 + 1.80i)T^{2} \)
29 \( 1 + (1.02 + 0.656i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-1.47 - 0.283i)T + (28.7 + 11.5i)T^{2} \)
37 \( 1 + (2.16 + 1.70i)T + (8.72 + 35.9i)T^{2} \)
41 \( 1 + (0.172 + 1.20i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.16 - 1.34i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (-0.751 + 1.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.5 - 1.00i)T + (52.0 - 10.0i)T^{2} \)
59 \( 1 + (-3.76 + 5.29i)T + (-19.2 - 55.7i)T^{2} \)
61 \( 1 + (-4.51 - 13.0i)T + (-47.9 + 37.7i)T^{2} \)
67 \( 1 + (-1.95 - 1.86i)T + (3.18 + 66.9i)T^{2} \)
71 \( 1 + (0.831 - 0.244i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-7.83 - 4.03i)T + (42.3 + 59.4i)T^{2} \)
79 \( 1 + (-2.53 - 0.241i)T + (77.5 + 14.9i)T^{2} \)
83 \( 1 + (-1.89 + 13.1i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-7.83 + 1.51i)T + (82.6 - 33.0i)T^{2} \)
97 \( 1 + (1.22 + 8.49i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.34289185849182840680784718313, −10.30221691757323655566930104641, −9.452017042315882414985626629067, −8.694363622480760347090820251063, −7.56044072545201738000431302159, −6.55942965076431080282816323809, −5.76335215134525625005090154031, −5.01754372908692261575617733931, −3.11638681337972616669582957184, −2.27295514830315210567879995184, 1.14322970682566468767871887867, 2.03949536966326350254158263063, 3.53695104490334086294412397760, 5.07611210836073753235186058969, 6.18213300408043243081778936040, 6.80707373328227514325695246957, 7.88660364148231967253816205344, 9.131748954717664963582238404599, 9.963880092684400206775833477852, 10.77241300689032070558423164366

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