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Properties

Label 2-648-9.4-c1-0-5
Degree $2$
Conductor $648$
Sign $0.939 + 0.342i$
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  + (−0.5 + 0.866i)5-s + (−2 − 3.46i)11-s + (2.5 − 4.33i)13-s + 5·17-s + 8·19-s + (−2 + 3.46i)23-s + (2 + 3.46i)25-s + (1.5 + 2.59i)29-s + (2 − 3.46i)31-s + 3·37-s + (−3 + 5.19i)41-s + (−2 − 3.46i)43-s + (−6 − 10.3i)47-s + (3.5 − 6.06i)49-s + 10·53-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.603 − 1.04i)11-s + (0.693 − 1.20i)13-s + 1.21·17-s + 1.83·19-s + (−0.417 + 0.722i)23-s + (0.400 + 0.692i)25-s + (0.278 + 0.482i)29-s + (0.359 − 0.622i)31-s + 0.493·37-s + (−0.468 + 0.811i)41-s + (−0.304 − 0.528i)43-s + (−0.875 − 1.51i)47-s + (0.5 − 0.866i)49-s + 1.37·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
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: \((2,\ 648,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46089 - 0.257595i\)
\(L(\frac12)\) \(\approx\) \(1.46089 - 0.257595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
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 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
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

−10.40669483059949278833418022701, −9.879597909853462448629663165980, −8.596916079704621307791166633108, −7.894018470313576269423089995058, −7.13372946002880008863686182679, −5.70045620935597034959393437541, −5.37295755941187757784743937648, −3.50975622336588429877136245341, −3.06609804457042375395309974160, −1.00238609440448937145581082726, 1.30779297828782502194162460003, 2.83860846897657684436331409330, 4.17236036404640762973616066106, 4.99882751558832397554317332751, 6.09586392461474002859707144693, 7.19773138232367590673549994069, 7.925724044080088551348835780880, 8.901119031133583659195871907494, 9.788968120591627280959432918501, 10.43719294184156948724130510520

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