Properties

Label 2-6e4-36.11-c1-0-7
Degree $2$
Conductor $1296$
Sign $0.642 - 0.766i$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
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  + (−2.59 + 1.5i)5-s + (1.5 + 0.866i)7-s + (2.59 − 4.5i)11-s + (1 + 1.73i)13-s − 6i·17-s + 6.92i·19-s + (2 − 3.46i)25-s + (5.19 + 3i)29-s + (−4.5 + 2.59i)31-s − 5.19·35-s + 8·37-s + (9 + 5.19i)43-s + (−5.19 + 9i)47-s + (−2 − 3.46i)49-s + 9i·53-s + ⋯
L(s)  = 1  + (−1.16 + 0.670i)5-s + (0.566 + 0.327i)7-s + (0.783 − 1.35i)11-s + (0.277 + 0.480i)13-s − 1.45i·17-s + 1.58i·19-s + (0.400 − 0.692i)25-s + (0.964 + 0.557i)29-s + (−0.808 + 0.466i)31-s − 0.878·35-s + 1.31·37-s + (1.37 + 0.792i)43-s + (−0.757 + 1.31i)47-s + (−0.285 − 0.494i)49-s + 1.23i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.416392210\)
\(L(\frac12)\) \(\approx\) \(1.416392210\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 1.5i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 4.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.19 - 3i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9 - 5.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.19 - 9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-3 - 1.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.59 - 4.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.680001097148985869706667048995, −8.833182783497748143984300023802, −8.100335247024714536429550989036, −7.44156674454193727745148065160, −6.50131028778037218974296443250, −5.68708378912370701899577346205, −4.47561157012033373488937011677, −3.63696410844035650314454850818, −2.81842519478783153527464956804, −1.12443652000872634148502462499, 0.74732653495368762674871241506, 2.09206890706172165020203639615, 3.73892848869018953152738225787, 4.34210113909531939192125059343, 5.02667837016721112678815387588, 6.35739486224535735435863501959, 7.24660602506422023024039938194, 7.963664363878880768181231291354, 8.595106609333800607308841544027, 9.447359460846758712226782149536

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