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Properties

Label 2-1476-1476.655-c0-0-2
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: $\chi_{1476} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1476,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6750350057\)
\(L(\frac12)\) \(\approx\) \(0.6750350057\)
\(L(1)\) 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

−9.714008343279663434762353918137, −9.190633045298207875504583577786, −8.275482905231870899260251054034, −7.18130842773441426701349395587, −6.45069295848700092396744254478, −5.92211204976561457382205366633, −5.50466367234330377659780636420, −4.12406262268698110371528353115, −2.83579259763322626380934522541, −1.12298435308916506085906130146, 1.01001426312663003231394820109, 1.85746359345177372809989674830, 3.58973078741846889713141160764, 4.51864643819025586621467443815, 5.00717774922590034762121783230, 6.38439877524365556147205972986, 7.14292898511342585988592504473, 7.86779541967494747483088352404, 9.273105306110083069423978064498, 9.647954219770647947725950019328

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