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Properties

Label 2-1476-1476.1147-c0-0-5
Degree $2$
Conductor $1476$
Sign $0.173 + 0.984i$
Analytic cond. $0.736619$
Root an. cond. $0.858265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + 9-s − 0.999·10-s + (−1 − 1.73i)11-s + (−0.499 + 0.866i)12-s + (0.499 − 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + 9-s − 0.999·10-s + (−1 − 1.73i)11-s + (−0.499 + 0.866i)12-s + (0.499 − 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(0.736619\)
Root analytic conductor: \(0.858265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (1147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1476,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.269447537\)
\(L(\frac12)\) \(\approx\) \(1.269447537\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - T \)
41 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 2T + T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.268644910146842581304808520052, −8.800073083613172759744438001728, −8.252923905727239576752200143767, −7.72634811054817307348826255651, −6.13994456167848882732450426388, −5.16020578303014785776161859702, −4.27912388403059770962778104930, −3.04747995251353979527698997043, −2.40623682513382176772668712794, −1.26291410200319182373913809526, 1.74905797969713757496601520973, 2.59886832209306016413304130508, 4.22281957197685316028017082396, 4.71743425635488370197878838227, 6.06046710155956419848652611576, 7.05209590870693509116555241202, 7.41260571836202307387823896206, 8.061525599745425706260764152946, 9.028596381768058299065497964343, 9.852093425342553834941792903900

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