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Properties

Label 2-483-161.4-c1-0-9
Degree $2$
Conductor $483$
Sign $-0.255 - 0.966i$
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  + (0.327 − 0.460i)2-s + (−0.235 + 0.971i)3-s + (0.549 + 1.58i)4-s + (−0.198 + 4.17i)5-s + (0.369 + 0.426i)6-s + (2.54 + 0.738i)7-s + (1.99 + 0.585i)8-s + (−0.888 − 0.458i)9-s + (1.85 + 1.45i)10-s + (−3.13 − 4.40i)11-s + (−1.67 + 0.159i)12-s + (0.0787 − 0.547i)13-s + (1.17 − 0.926i)14-s + (−4.00 − 1.17i)15-s + (−1.71 + 1.35i)16-s + (−1.13 + 0.219i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.325i)2-s + (−0.136 + 0.561i)3-s + (0.274 + 0.794i)4-s + (−0.0889 + 1.86i)5-s + (0.150 + 0.174i)6-s + (0.960 + 0.279i)7-s + (0.705 + 0.207i)8-s + (−0.296 − 0.152i)9-s + (0.586 + 0.461i)10-s + (−0.946 − 1.32i)11-s + (−0.483 + 0.0461i)12-s + (0.0218 − 0.151i)13-s + (0.313 − 0.247i)14-s + (−1.03 − 0.303i)15-s + (−0.429 + 0.338i)16-s + (−0.276 + 0.0532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.255 - 0.966i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.255 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02197 + 1.32649i\)
\(L(\frac12)\) \(\approx\) \(1.02197 + 1.32649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (-2.54 - 0.738i)T \)
23 \( 1 + (-3.87 + 2.82i)T \)
good2 \( 1 + (-0.327 + 0.460i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (0.198 - 4.17i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (3.13 + 4.40i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (-0.0787 + 0.547i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.13 - 0.219i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (-3.13 - 0.605i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (2.55 + 2.94i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-3.98 + 3.80i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (-8.01 - 4.12i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (-2.12 - 1.36i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-0.539 + 0.158i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-3.26 - 5.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (12.0 + 4.84i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (0.280 + 0.220i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (2.36 + 9.74i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (-5.66 - 0.541i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (-4.93 - 10.8i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-2.79 - 8.08i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (7.82 - 3.13i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (12.0 - 7.73i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-1.39 - 1.33i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (-8.34 - 5.36i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.14518069853191940610430343024, −10.85754179110140695132247579387, −9.745850758874196455102959556751, −8.204928371240376995578638115826, −7.81100772760447201256950882536, −6.63319491103454572275548524671, −5.60808175075464898039902321622, −4.26134452091224374007728318601, −3.08178432335519814497266578201, −2.58832243670118253284841885800, 1.03813014014035838001265481653, 1.96672529800461685876495755951, 4.58427862316011856797783271939, 4.94256147085616132159921493457, 5.75986274204645589888901856288, 7.28565196020678133000481860214, 7.74108185388470498432919033663, 8.900797157661527714387065067334, 9.722738750995847825987287654052, 10.82989069198720012929470387233

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