Properties

Label 2-648-648.155-c1-0-46
Degree $2$
Conductor $648$
Sign $-0.421 - 0.906i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.686 + 1.23i)2-s + (−1.64 + 0.537i)3-s + (−1.05 + 1.69i)4-s + (1.64 − 1.08i)5-s + (−1.79 − 1.66i)6-s + (3.20 + 3.02i)7-s + (−2.82 − 0.142i)8-s + (2.42 − 1.76i)9-s + (2.46 + 1.29i)10-s + (0.555 + 1.10i)11-s + (0.830 − 3.36i)12-s + (5.40 − 2.32i)13-s + (−1.53 + 6.03i)14-s + (−2.12 + 2.66i)15-s + (−1.76 − 3.59i)16-s + (−0.839 − 1.00i)17-s + ⋯
L(s)  = 1  + (0.485 + 0.874i)2-s + (−0.950 + 0.310i)3-s + (−0.528 + 0.848i)4-s + (0.736 − 0.484i)5-s + (−0.732 − 0.680i)6-s + (1.21 + 1.14i)7-s + (−0.998 − 0.0504i)8-s + (0.807 − 0.589i)9-s + (0.780 + 0.408i)10-s + (0.167 + 0.333i)11-s + (0.239 − 0.970i)12-s + (1.49 − 0.646i)13-s + (−0.411 + 1.61i)14-s + (−0.549 + 0.688i)15-s + (−0.440 − 0.897i)16-s + (−0.203 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.421 - 0.906i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.421 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.944789 + 1.48082i\)
\(L(\frac12)\) \(\approx\) \(0.944789 + 1.48082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.686 - 1.23i)T \)
3 \( 1 + (1.64 - 0.537i)T \)
good5 \( 1 + (-1.64 + 1.08i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (-3.20 - 3.02i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (-0.555 - 1.10i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (-5.40 + 2.32i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (0.839 + 1.00i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-0.869 - 0.729i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-2.24 - 2.38i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (9.66 - 1.12i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (-0.734 - 3.09i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (4.78 - 0.843i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (-8.90 - 6.63i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (-0.456 + 7.82i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (9.28 + 2.20i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (3.53 + 6.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.88 + 5.74i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (-7.14 + 2.13i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (-6.40 - 0.748i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (-1.77 - 0.646i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.89 - 2.87i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (5.76 - 4.29i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (7.58 - 5.64i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (4.69 + 12.9i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-9.18 - 6.04i)T + (38.4 + 89.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.23412629759110557096612057795, −9.757573865680990824477811646901, −8.965566119305541812001365996644, −8.260858509638953208109538758913, −7.09270836436932762075662713944, −5.92289694792713650408442418465, −5.48112239604662335221585187962, −4.87693287465153530944928507321, −3.59895958086704455392180859050, −1.62331148288848681920510898240, 1.08925313066468543115343805339, 1.99047919453574613464998753409, 3.82937418254794547887674462176, 4.58730363108688642072893774261, 5.73629075698388342897575344077, 6.37005592979472960620605693450, 7.44877078637712271856589499797, 8.702152945207710218920438026648, 9.839055828454065873018940016298, 10.74451947553635687733356100679

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