Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.838 + 0.544i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.34 − 4.63i)3-s + (−5.19 − 4.35i)5-s + (−17.8 − 6.47i)7-s + (−16.0 − 21.7i)9-s + (−0.134 + 0.112i)11-s + (−1.24 + 7.04i)13-s + (−32.3 + 13.8i)15-s + (−26.6 + 46.0i)17-s + (−65.4 − 113. i)19-s + (−71.7 + 67.4i)21-s + (129. − 46.9i)23-s + (−13.7 − 77.8i)25-s + (−138. + 23.5i)27-s + (−9.32 − 52.8i)29-s + (139. − 50.6i)31-s + ⋯
L(s)  = 1  + (0.450 − 0.892i)3-s + (−0.464 − 0.389i)5-s + (−0.961 − 0.349i)7-s + (−0.594 − 0.804i)9-s + (−0.00367 + 0.00308i)11-s + (−0.0265 + 0.150i)13-s + (−0.557 + 0.239i)15-s + (−0.379 + 0.657i)17-s + (−0.790 − 1.36i)19-s + (−0.745 + 0.700i)21-s + (1.17 − 0.426i)23-s + (−0.109 − 0.622i)25-s + (−0.985 + 0.167i)27-s + (−0.0597 − 0.338i)29-s + (0.806 − 0.293i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.838 + 0.544i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.838 + 0.544i)\)
\(L(2)\)  \(\approx\)  \(0.305540 - 1.03138i\)
\(L(\frac12)\)  \(\approx\)  \(0.305540 - 1.03138i\)
\(L(\frac{5}{2})\)   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 \( 1 \)
3 \( 1 + (-2.34 + 4.63i)T \)
good5 \( 1 + (5.19 + 4.35i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (17.8 + 6.47i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (0.134 - 0.112i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (1.24 - 7.04i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (26.6 - 46.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (65.4 + 113. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-129. + 46.9i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (9.32 + 52.8i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (-139. + 50.6i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (-58.6 + 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (27.9 - 158. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (-51.8 + 43.5i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (-597. - 217. i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 - 36.5T + 1.48e5T^{2} \)
59 \( 1 + (-574. - 482. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (64.9 + 23.6i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (-109. + 622. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (66.9 - 116. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (435. + 754. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-55.7 - 315. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (197. + 1.12e3i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (259. + 449. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.15e3 - 968. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.95893730241121448781908631863, −12.02490789100296310659717053898, −10.77097670386416433388873057792, −9.280808728401417617089632730753, −8.416202751812204239956251613708, −7.14390172120894965556186266143, −6.26483240446214220401082640484, −4.25483257132516407592158115150, −2.67340487711749434398017581750, −0.54924009681971177603435860525, 2.82286704886462681029135212250, 3.93092244929222327765133042337, 5.51232886296881865832646123001, 7.01094363067852154210803417699, 8.398350596516054878067164701138, 9.424440838750729693266206480408, 10.36420854464652644073823568912, 11.39212527010611592988567705728, 12.64188935281719571313603869763, 13.76245839891706451557218155922

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