Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.991 + 0.129i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.253 + 5.65i)2-s + (−31.8 − 2.86i)4-s + (−84.3 + 48.6i)5-s + (−87.9 − 50.7i)7-s + (24.2 − 179. i)8-s + (−253. − 488. i)10-s + (73.2 − 126. i)11-s + (402. + 697. i)13-s + (309. − 484. i)14-s + (1.00e3 + 182. i)16-s + 1.23e3i·17-s − 2.61e3i·19-s + (2.82e3 − 1.30e3i)20-s + (698. + 446. i)22-s + (−84.5 − 146. i)23-s + ⋯
L(s)  = 1  + (−0.0448 + 0.998i)2-s + (−0.995 − 0.0895i)4-s + (−1.50 + 0.870i)5-s + (−0.678 − 0.391i)7-s + (0.134 − 0.990i)8-s + (−0.802 − 1.54i)10-s + (0.182 − 0.316i)11-s + (0.661 + 1.14i)13-s + (0.421 − 0.660i)14-s + (0.983 + 0.178i)16-s + 1.03i·17-s − 1.66i·19-s + (1.58 − 0.732i)20-s + (0.307 + 0.196i)22-s + (−0.0333 − 0.0577i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.991 + 0.129i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.991 + 0.129i)\)
\(L(3)\)  \(\approx\)  \(0.574971 - 0.0373996i\)
\(L(\frac12)\)  \(\approx\)  \(0.574971 - 0.0373996i\)
\(L(\frac{7}{2})\)   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 \( 1 + (0.253 - 5.65i)T \)
3 \( 1 \)
good5 \( 1 + (84.3 - 48.6i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (87.9 + 50.7i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-73.2 + 126. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-402. - 697. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.23e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.61e3iT - 2.47e6T^{2} \)
23 \( 1 + (84.5 + 146. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (1.95e3 + 1.12e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.78e3 + 1.03e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 445.T + 6.93e7T^{2} \)
41 \( 1 + (-1.72e3 + 993. i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-3.46e3 - 2.00e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-3.53e3 + 6.13e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 6.23e3iT - 4.18e8T^{2} \)
59 \( 1 + (1.67e4 + 2.89e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.47e4 + 2.54e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.33e4 + 1.34e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.63e4T + 1.80e9T^{2} \)
73 \( 1 + 3.82e4T + 2.07e9T^{2} \)
79 \( 1 + (-5.45e4 - 3.14e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.99e4 + 6.92e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 9.66e4iT - 5.58e9T^{2} \)
97 \( 1 + (8.31e3 - 1.44e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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

−12.97512685845699711194334639599, −11.60311627129471621600130240683, −10.67408845561589638303740032237, −9.205005163312395002669107302990, −8.129171088410518602096680555531, −7.02027167921904117908556059869, −6.38731198223517103831589886984, −4.34665430793949531399583841244, −3.48385937837794462308188778217, −0.30138482399323671769551012651, 0.986480116068034184803379712771, 3.16252969864550391197768295760, 4.16204691836585904563818881732, 5.52314072929022057328995710576, 7.68426445885241249986458794790, 8.538749432278357752861953375294, 9.595797173299715784303688478596, 10.83688573539948730364687242101, 12.00507645427673332747654956818, 12.40753435473484752679996043213

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