Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.852 - 0.522i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.49 − 1.95i)2-s + (8.36 + 13.6i)4-s − 47.1·5-s − 67.4i·7-s + (−2.56 − 63.9i)8-s + (164. + 92.0i)10-s + 69.3i·11-s + 3.28·13-s + (−131. + 235. i)14-s + (−115. + 228. i)16-s + 116.·17-s + 513. i·19-s + (−394. − 642. i)20-s + (135. − 242. i)22-s + 134. i·23-s + ⋯
L(s)  = 1  + (−0.872 − 0.488i)2-s + (0.522 + 0.852i)4-s − 1.88·5-s − 1.37i·7-s + (−0.0400 − 0.999i)8-s + (1.64 + 0.920i)10-s + 0.573i·11-s + 0.0194·13-s + (−0.671 + 1.20i)14-s + (−0.453 + 0.891i)16-s + 0.404·17-s + 1.42i·19-s + (−0.985 − 1.60i)20-s + (0.280 − 0.500i)22-s + 0.253i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.852 - 0.522i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.852 - 0.522i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.526598 + 0.148672i\)
\(L(\frac12)\)  \(\approx\)  \(0.526598 + 0.148672i\)
\(L(3)\)   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 \( 1 + (3.49 + 1.95i)T \)
3 \( 1 \)
good5 \( 1 + 47.1T + 625T^{2} \)
7 \( 1 + 67.4iT - 2.40e3T^{2} \)
11 \( 1 - 69.3iT - 1.46e4T^{2} \)
13 \( 1 - 3.28T + 2.85e4T^{2} \)
17 \( 1 - 116.T + 8.35e4T^{2} \)
19 \( 1 - 513. iT - 1.30e5T^{2} \)
23 \( 1 - 134. iT - 2.79e5T^{2} \)
29 \( 1 - 772.T + 7.07e5T^{2} \)
31 \( 1 - 1.06e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.03e3T + 1.87e6T^{2} \)
41 \( 1 - 2.27e3T + 2.82e6T^{2} \)
43 \( 1 + 2.52e3iT - 3.41e6T^{2} \)
47 \( 1 + 694. iT - 4.87e6T^{2} \)
53 \( 1 + 1.16e3T + 7.89e6T^{2} \)
59 \( 1 - 5.53e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.20e3T + 1.38e7T^{2} \)
67 \( 1 - 5.36e3iT - 2.01e7T^{2} \)
71 \( 1 + 166. iT - 2.54e7T^{2} \)
73 \( 1 - 5.08e3T + 2.83e7T^{2} \)
79 \( 1 + 724. iT - 3.89e7T^{2} \)
83 \( 1 - 8.15e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.96e3T + 6.27e7T^{2} \)
97 \( 1 + 278.T + 8.85e7T^{2} \)
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

−12.52881219452139956335626252635, −11.97286211339553864675171657440, −10.83407928114929195997003137672, −10.16535876718558111156839645690, −8.543083326610122583623725893187, −7.62669459374345515732745942875, −7.04554169162723109940126977758, −4.26419859592640515379828359854, −3.45600826002088788163322174955, −0.990339888930384510456613000715, 0.43565287824892026175009391459, 2.88197797315220148751088093545, 4.82974098136142568435668596757, 6.32948707739414540400014675815, 7.65004301491122218059735604048, 8.430600102213332260073531983841, 9.272488132691715564807081962841, 11.00385586263974718500719165523, 11.57635846210571398354744203555, 12.53604126132410178225878501830

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