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Properties

Label 2-648-9.4-c1-0-9
Degree $2$
Conductor $648$
Sign $0.173 + 0.984i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  + (1 − 1.73i)5-s + (−2 − 3.46i)11-s + (1 − 1.73i)13-s + 2·17-s − 4·19-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s + (−3 − 5.19i)29-s + (−4 + 6.92i)31-s + 6·37-s + (3 − 5.19i)41-s + (−2 − 3.46i)43-s + (3.5 − 6.06i)49-s − 2·53-s − 7.99·55-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (−0.603 − 1.04i)11-s + (0.277 − 0.480i)13-s + 0.485·17-s − 0.917·19-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.557 − 0.964i)29-s + (−0.718 + 1.24i)31-s + 0.986·37-s + (0.468 − 0.811i)41-s + (−0.304 − 0.528i)43-s + (0.5 − 0.866i)49-s − 0.274·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10132 - 0.924121i\)
\(L(\frac12)\) \(\approx\) \(1.10132 - 0.924121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.55148577808006058343086342745, −9.360380108516642797883713940191, −8.624007881627953683019060061162, −7.980359729402398523721405884329, −6.69299185586611711619627501552, −5.69395885640079870006386771579, −5.02244930777681257047983776151, −3.71741783398813422121774248189, −2.43903165434531482004299607705, −0.795445496484644195298392014606, 1.81482394597607346601195346320, 2.93723664021228436927822728302, 4.21186959740816363360918420146, 5.35184931790300732064915431403, 6.34008160466344299310352664117, 7.20850623489491585416891420720, 7.959977925105450543168745352543, 9.296350878961516544050763838765, 9.800694521284593924296395428975, 10.82910976815498650411108649147

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