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Properties

Label 2-648-72.11-c1-0-16
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + 2.82i·8-s + (2.44 + 1.41i)11-s + (−2.00 + 3.46i)16-s + 5.65i·17-s + 2·19-s + (1.99 + 3.46i)22-s + (2.5 − 4.33i)25-s + (−4.89 + 2.82i)32-s + (−4.00 + 6.92i)34-s + (2.44 + 1.41i)38-s + (−9.79 + 5.65i)41-s + (5 − 8.66i)43-s + 5.65i·44-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (0.738 + 0.426i)11-s + (−0.500 + 0.866i)16-s + 1.37i·17-s + 0.458·19-s + (0.426 + 0.738i)22-s + (0.5 − 0.866i)25-s + (−0.866 + 0.499i)32-s + (−0.685 + 1.18i)34-s + (0.397 + 0.229i)38-s + (−1.53 + 0.883i)41-s + (0.762 − 1.32i)43-s + 0.852i·44-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: $\chi_{648} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91875 + 1.61002i\)
\(L(\frac12)\) \(\approx\) \(1.91875 + 1.61002i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.44 - 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (9.79 - 5.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-12.2 + 7.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.44 - 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.89401776483396621121597160293, −9.953994069293537346548026229012, −8.747868538769108074589700704759, −8.051273100514968163496992480913, −6.94610833045269602799251803276, −6.31847905302640599106477930363, −5.27082070028871773508167765680, −4.26228403567267934635315587836, −3.38007721658756800872919200214, −1.91944243790322267206906112850, 1.15949047844928478056530366707, 2.70939775202098723860756402694, 3.66233335730430501892377238223, 4.78967384552911708289730842874, 5.62950742355998522429812203905, 6.67421754769981061490747647340, 7.44547345673844390365579788830, 8.902816311631928154592579738551, 9.600181588161216776049823954034, 10.54625879007997392541505825832

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