↑

Properties

Label 2-6e3-9.2-c2-0-3
Degree $2$
Conductor $216$
Sign $0.603 + 0.797i$
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  + (0.0440 − 0.0254i)5-s + (4.52 − 7.84i)7-s + (−3.29 − 1.90i)11-s + (0.216 + 0.375i)13-s − 26.2i·17-s + 34.2·19-s + (29.9 − 17.3i)23-s + (−12.4 + 21.6i)25-s + (−14.0 − 8.10i)29-s + (−17.1 − 29.7i)31-s − 0.460i·35-s + 29.2·37-s + (−48.7 + 28.1i)41-s + (−3.94 + 6.83i)43-s + (33.4 + 19.3i)47-s + ⋯
L(s)  = 1  + (0.00880 − 0.00508i)5-s + (0.647 − 1.12i)7-s + (−0.299 − 0.172i)11-s + (0.0166 + 0.0288i)13-s − 1.54i·17-s + 1.80·19-s + (1.30 − 0.752i)23-s + (−0.499 + 0.865i)25-s + (−0.483 − 0.279i)29-s + (−0.553 − 0.959i)31-s − 0.0131i·35-s + 0.791·37-s + (−1.18 + 0.685i)41-s + (−0.0917 + 0.158i)43-s + (0.711 + 0.410i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.603 + 0.797i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42372 - 0.708015i\)
\(L(\frac12)\) \(\approx\) \(1.42372 - 0.708015i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.0440 + 0.0254i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.52 + 7.84i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (3.29 + 1.90i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.216 - 0.375i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 26.2iT - 289T^{2} \)
19 \( 1 - 34.2T + 361T^{2} \)
23 \( 1 + (-29.9 + 17.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (14.0 + 8.10i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (17.1 + 29.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 29.2T + 1.36e3T^{2} \)
41 \( 1 + (48.7 - 28.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (3.94 - 6.83i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-33.4 - 19.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 50.5iT - 2.80e3T^{2} \)
59 \( 1 + (-8.54 + 4.93i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (36.5 - 63.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (12.6 + 21.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 97.8iT - 5.04e3T^{2} \)
73 \( 1 + 77.0T + 5.32e3T^{2} \)
79 \( 1 + (42.1 - 72.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (40.6 + 23.4i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 + (32.4 - 56.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−11.64166653462482130015372612010, −11.16892017253011616202388744033, −9.994016370970845996116237757524, −9.103051895612425925461105955159, −7.62533938782479897555235922883, −7.21337823628303826398339352794, −5.51411593097828802737726511448, −4.49341275933664568170475580854, −3.02473180882254094199143850690, −0.989399795376746464926906543620, 1.76429894830359508876465030460, 3.33960968152408349756510724051, 5.02703927177714132515297135633, 5.81545952324260937857258169720, 7.28128768998186305071061415380, 8.330918558305479213339906044328, 9.181570987627718843877859403468, 10.32519573756307969531890030866, 11.39184342290847615382811956448, 12.15401910367854979681288553500

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