↑

Properties

Label 2-6e3-72.67-c2-0-2
Degree $2$
Conductor $216$
Sign $-0.707 + 0.706i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0392 + 1.99i)2-s + (−3.99 − 0.156i)4-s + (1.50 + 0.871i)5-s + (−7.93 + 4.58i)7-s + (0.470 − 7.98i)8-s + (−1.80 + 2.98i)10-s + (4.09 + 7.08i)11-s + (−19.9 − 11.5i)13-s + (−8.85 − 16.0i)14-s + (15.9 + 1.25i)16-s − 18.4·17-s − 7.06·19-s + (−5.89 − 3.72i)20-s + (−14.3 + 7.90i)22-s + (9.33 + 5.39i)23-s + ⋯
L(s)  = 1  + (−0.0196 + 0.999i)2-s + (−0.999 − 0.0392i)4-s + (0.301 + 0.174i)5-s + (−1.13 + 0.654i)7-s + (0.0587 − 0.998i)8-s + (−0.180 + 0.298i)10-s + (0.371 + 0.644i)11-s + (−1.53 − 0.885i)13-s + (−0.632 − 1.14i)14-s + (0.996 + 0.0783i)16-s − 1.08·17-s − 0.371·19-s + (−0.294 − 0.186i)20-s + (−0.651 + 0.359i)22-s + (0.405 + 0.234i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.707 + 0.706i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.707 + 0.706i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.140271 - 0.338846i\)
\(L(\frac12)\) \(\approx\) \(0.140271 - 0.338846i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0392 - 1.99i)T \)
3 \( 1 \)
good5 \( 1 + (-1.50 - 0.871i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (7.93 - 4.58i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-4.09 - 7.08i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (19.9 + 11.5i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 18.4T + 289T^{2} \)
19 \( 1 + 7.06T + 361T^{2} \)
23 \( 1 + (-9.33 - 5.39i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (14.3 - 8.28i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.18 - 0.684i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 59.1iT - 1.36e3T^{2} \)
41 \( 1 + (-19.6 + 34.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-8.49 - 14.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-25.4 + 14.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 49.1iT - 2.80e3T^{2} \)
59 \( 1 + (32.8 - 56.8i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (11.8 - 6.86i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (24.8 - 43.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 136. iT - 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 + (-82.5 + 47.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-39.2 - 67.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 72.9T + 7.92e3T^{2} \)
97 \( 1 + (25.9 + 44.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−12.79090840883975508696771009618, −12.13132905846731650468754735248, −10.31990733020262612556793717360, −9.611349857492024734377892322046, −8.805720147904420455719978571362, −7.45123266978057739804991068128, −6.60742841541465549378600075880, −5.64656540882615163552868750774, −4.43004940500286715092132067051, −2.72190228153671769155026623654, 0.19314740739809117478537727051, 2.20181125871244267485149218060, 3.62759290292616909274898364345, 4.73213954812854477364662552030, 6.24624964205513827341332575054, 7.45388567434191631038754303782, 9.113159034311058141794085642491, 9.458443745407727353703834335401, 10.55468385217965812178517812189, 11.44047537859955453137558677504

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