Properties

Label 2-483-161.18-c1-0-5
Degree $2$
Conductor $483$
Sign $0.784 + 0.619i$
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  + (−0.638 − 2.63i)2-s + (−0.327 − 0.945i)3-s + (−4.73 + 2.44i)4-s + (−0.0914 + 0.0366i)5-s + (−2.27 + 1.46i)6-s + (−1.45 + 2.20i)7-s + (5.90 + 6.81i)8-s + (−0.786 + 0.618i)9-s + (0.154 + 0.217i)10-s + (−1.01 + 4.17i)11-s + (3.85 + 3.67i)12-s + (2.68 − 5.87i)13-s + (6.73 + 2.43i)14-s + (0.0645 + 0.0744i)15-s + (7.97 − 11.2i)16-s + (0.164 + 3.45i)17-s + ⋯
L(s)  = 1  + (−0.451 − 1.86i)2-s + (−0.188 − 0.545i)3-s + (−2.36 + 1.22i)4-s + (−0.0409 + 0.0163i)5-s + (−0.929 + 0.597i)6-s + (−0.551 + 0.834i)7-s + (2.08 + 2.40i)8-s + (−0.262 + 0.206i)9-s + (0.0489 + 0.0687i)10-s + (−0.305 + 1.25i)11-s + (1.11 + 1.06i)12-s + (0.744 − 1.62i)13-s + (1.80 + 0.649i)14-s + (0.0166 + 0.0192i)15-s + (1.99 − 2.80i)16-s + (0.0399 + 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.784 + 0.619i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.784 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557258 - 0.193459i\)
\(L(\frac12)\) \(\approx\) \(0.557258 - 0.193459i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 2.20i)T \)
23 \( 1 + (0.209 - 4.79i)T \)
good2 \( 1 + (0.638 + 2.63i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (0.0914 - 0.0366i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (1.01 - 4.17i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (-2.68 + 5.87i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.164 - 3.45i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.00286 + 0.0600i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (-4.91 + 3.15i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-2.01 + 0.388i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (0.876 - 0.689i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (1.14 - 7.98i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (7.14 - 8.25i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (-5.45 - 9.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.72 - 0.260i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (3.58 + 5.03i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (0.569 - 1.64i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (-7.82 + 7.45i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (4.36 + 1.28i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.94 - 1.51i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (2.97 - 0.284i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (-1.54 - 10.7i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-0.297 - 0.0573i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (1.53 - 10.6i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.94078013708708376314136635139, −10.04198633809247119276154895240, −9.523576032010842572147241151677, −8.318481265379397952163121005672, −7.79008937045168725851604201316, −6.04801910768782231462261565547, −4.91567924392931052946105548366, −3.50539930617258034972135023298, −2.59878536767688988124906174612, −1.38965061592127132541390435890, 0.47402238926178426649130055929, 3.74780951171443784815230854385, 4.61274758331489210543413825597, 5.74795386507759372549200750605, 6.57428629679210376824362788008, 7.18317616456121613032101749988, 8.570500278148995540467841521021, 8.830102308402820740984770812798, 9.993214183456929362600796000129, 10.60044628884469338726339881742

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