↑

Properties

Label 2-6e3-72.43-c2-0-10
Degree $2$
Conductor $216$
Sign $0.978 + 0.207i$
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  + (−1.75 − 0.959i)2-s + (2.15 + 3.36i)4-s + (6.07 − 3.50i)5-s + (8.07 + 4.66i)7-s + (−0.553 − 7.98i)8-s + (−14.0 + 0.323i)10-s + (−8.69 + 15.0i)11-s + (−3.23 + 1.86i)13-s + (−9.69 − 15.9i)14-s + (−6.68 + 14.5i)16-s + 9.68·17-s + 21.4·19-s + (24.9 + 12.8i)20-s + (29.6 − 18.0i)22-s + (2.94 − 1.69i)23-s + ⋯
L(s)  = 1  + (−0.877 − 0.479i)2-s + (0.539 + 0.842i)4-s + (1.21 − 0.701i)5-s + (1.15 + 0.665i)7-s + (−0.0691 − 0.997i)8-s + (−1.40 + 0.0323i)10-s + (−0.790 + 1.36i)11-s + (−0.248 + 0.143i)13-s + (−0.692 − 1.13i)14-s + (−0.418 + 0.908i)16-s + 0.569·17-s + 1.12·19-s + (1.24 + 0.644i)20-s + (1.34 − 0.821i)22-s + (0.127 − 0.0738i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.978 + 0.207i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37846 - 0.144479i\)
\(L(\frac12)\) \(\approx\) \(1.37846 - 0.144479i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.75 + 0.959i)T \)
3 \( 1 \)
good5 \( 1 + (-6.07 + 3.50i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-8.07 - 4.66i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.69 - 15.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (3.23 - 1.86i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 9.68T + 289T^{2} \)
19 \( 1 - 21.4T + 361T^{2} \)
23 \( 1 + (-2.94 + 1.69i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (18.7 + 10.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-20.8 + 12.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 26.1iT - 1.36e3T^{2} \)
41 \( 1 + (-15.6 - 27.1i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.01 + 5.22i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-39.7 - 22.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 17.8iT - 2.80e3T^{2} \)
59 \( 1 + (26.8 + 46.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (73.5 + 42.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (9.38 + 16.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 44.5iT - 5.04e3T^{2} \)
73 \( 1 - 35.5T + 5.32e3T^{2} \)
79 \( 1 + (66.2 + 38.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (45.5 - 78.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 31.0T + 7.92e3T^{2} \)
97 \( 1 + (21.4 - 37.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−12.06949558262753259417472156023, −10.98635384943086788577224772227, −9.784670245846757296664749496690, −9.425526333904647962451703077169, −8.203382888022527627456578890325, −7.37932364315325097320551730511, −5.69765230958013927630016985153, −4.71566572831285230519905128839, −2.45324281550810839253878276174, −1.52282690130564426320495201457, 1.24705788365947140047082701804, 2.82056700887640427697777657591, 5.22012175315581908575714484943, 5.94402851458309618509337065271, 7.25969821418432233215974546592, 8.031799157347142853849055214633, 9.176028760091913149248993176022, 10.33157501583640451458473529675, 10.70794386983432576218008693300, 11.69993643264925516637589886216

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