Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.900 - 0.434i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.68 − 1.56i)2-s + (11.1 + 11.5i)4-s + (−1.01 + 1.75i)5-s + (20.0 − 11.5i)7-s + (−22.8 − 59.7i)8-s + (6.48 − 4.88i)10-s + (−4.32 + 2.49i)11-s + (−137. + 238. i)13-s + (−91.8 + 11.2i)14-s + (−9.35 + 255. i)16-s + 266.·17-s − 367. i·19-s + (−31.5 + 7.83i)20-s + (19.8 − 2.42i)22-s + (544. + 314. i)23-s + ⋯
L(s)  = 1  + (−0.920 − 0.391i)2-s + (0.694 + 0.719i)4-s + (−0.0406 + 0.0703i)5-s + (0.408 − 0.236i)7-s + (−0.357 − 0.934i)8-s + (0.0648 − 0.0488i)10-s + (−0.0357 + 0.0206i)11-s + (−0.815 + 1.41i)13-s + (−0.468 + 0.0573i)14-s + (−0.0365 + 0.999i)16-s + 0.920·17-s − 1.01i·19-s + (−0.0788 + 0.0195i)20-s + (0.0410 − 0.00501i)22-s + (1.02 + 0.594i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.900 - 0.434i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.900 - 0.434i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.07275 + 0.245262i\)
\(L(\frac12)\)  \(\approx\)  \(1.07275 + 0.245262i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (3.68 + 1.56i)T \)
3 \( 1 \)
good5 \( 1 + (1.01 - 1.75i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-20.0 + 11.5i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (4.32 - 2.49i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (137. - 238. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 266.T + 8.35e4T^{2} \)
19 \( 1 + 367. iT - 1.30e5T^{2} \)
23 \( 1 + (-544. - 314. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-319. - 553. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-1.19e3 - 687. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.46e3T + 1.87e6T^{2} \)
41 \( 1 + (593. - 1.02e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.43e3 + 825. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (307. - 177. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 5.29e3T + 7.89e6T^{2} \)
59 \( 1 + (-5.22e3 - 3.01e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (833. + 1.44e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (1.90e3 + 1.10e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 524. iT - 2.54e7T^{2} \)
73 \( 1 + 1.49e3T + 2.83e7T^{2} \)
79 \( 1 + (-4.44e3 + 2.56e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (6.91e3 - 3.99e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 8.86e3T + 6.27e7T^{2} \)
97 \( 1 + (3.40e3 + 5.90e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.79375814161301929220597496858, −11.71740758548666401961311015680, −10.99202614232278023631433990046, −9.748686801239933376831731514516, −8.921985504701647228523681292063, −7.59514199875557197568225539452, −6.73894259457453846046422527386, −4.71952512639696909448590351340, −2.94502735140006268417057326160, −1.27244621694711728187732417212, 0.77603793797390430330979597529, 2.67656354170817125303490563006, 5.03911184885412605977723575961, 6.20475879083290786807593261614, 7.71879623852163485325016978480, 8.310321654435023849685686813942, 9.767439152917441884811861686533, 10.47243220206186918480885527463, 11.73388284079631756126121446700, 12.72547407543930725013672760328

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