↑

Properties

Label 2-648-72.11-c1-0-15
Degree $2$
Conductor $648$
Sign $0.999 - 0.0209i$
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  + (−1.02 + 0.977i)2-s + (0.0884 − 1.99i)4-s + (0.252 + 0.437i)5-s + (−2.96 − 1.71i)7-s + (1.86 + 2.12i)8-s + (−0.686 − 0.200i)10-s + (2.87 + 1.65i)11-s + (−2.20 + 1.27i)13-s + (4.70 − 1.15i)14-s + (−3.98 − 0.353i)16-s − 5.04i·17-s + 4.74·19-s + (0.896 − 0.466i)20-s + (−4.55 + 1.11i)22-s + (3.22 + 5.57i)23-s + ⋯
L(s)  = 1  + (−0.722 + 0.691i)2-s + (0.0442 − 0.999i)4-s + (0.113 + 0.195i)5-s + (−1.12 − 0.647i)7-s + (0.658 + 0.752i)8-s + (−0.216 − 0.0633i)10-s + (0.866 + 0.500i)11-s + (−0.612 + 0.353i)13-s + (1.25 − 0.307i)14-s + (−0.996 − 0.0883i)16-s − 1.22i·17-s + 1.08·19-s + (0.200 − 0.104i)20-s + (−0.971 + 0.237i)22-s + (0.671 + 1.16i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.999 - 0.0209i$
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: \((2,\ 648,\ (\ :1/2),\ 0.999 - 0.0209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.927379 + 0.00970101i\)
\(L(\frac12)\) \(\approx\) \(0.927379 + 0.00970101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 - 0.977i)T \)
3 \( 1 \)
good5 \( 1 + (-0.252 - 0.437i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.96 + 1.71i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.87 - 1.65i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.20 - 1.27i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.04iT - 17T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
23 \( 1 + (-3.22 - 5.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.71 + 4.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.17 + 2.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 + (-1.62 + 0.939i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.43 - 9.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.93T + 53T^{2} \)
59 \( 1 + (-5.74 + 3.31i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.41 - 2.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 7.74T + 73T^{2} \)
79 \( 1 + (3.72 + 2.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.12 + 1.80i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.0iT - 89T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.10876207720406471816161153450, −9.611079653504700853327160739177, −9.115509827141622780199180641760, −7.64879705180226714320797316141, −7.06879402028829247997623553358, −6.42818292544464814244550086212, −5.28049171237777617333251174239, −4.10978161312832610093126534490, −2.61165013591004305779275422079, −0.794071797595524469053396231255, 1.14774585330418597374804208792, 2.79683202900332938432981422723, 3.49910188092475256161855497310, 4.94527941482272882867376760858, 6.31033036979428978256092656131, 7.00103979580291069541974087659, 8.379110883820714790831818410523, 8.851269638699697804656444919508, 9.773073656174297504360635672135, 10.31801064165750075996267964904

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