Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.146 - 0.989i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.95 − 2.27i)3-s + (−3.25 + 8.93i)5-s + (0.410 − 2.32i)7-s + (−1.38 + 8.89i)9-s + (4.40 + 12.1i)11-s + (−12.2 + 10.2i)13-s + (26.6 − 10.0i)15-s + (12.2 − 7.04i)17-s + (−3.29 + 5.70i)19-s + (−6.09 + 3.60i)21-s + (−25.9 + 4.58i)23-s + (−50.0 − 41.9i)25-s + (22.9 − 14.2i)27-s + (0.977 − 1.16i)29-s + (−0.620 − 3.52i)31-s + ⋯
L(s)  = 1  + (−0.650 − 0.759i)3-s + (−0.650 + 1.78i)5-s + (0.0585 − 0.332i)7-s + (−0.153 + 0.988i)9-s + (0.400 + 1.10i)11-s + (−0.939 + 0.788i)13-s + (1.77 − 0.668i)15-s + (0.717 − 0.414i)17-s + (−0.173 + 0.300i)19-s + (−0.290 + 0.171i)21-s + (−1.12 + 0.199i)23-s + (−2.00 − 1.67i)25-s + (0.850 − 0.526i)27-s + (0.0337 − 0.0401i)29-s + (−0.0200 − 0.113i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.146 - 0.989i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (65, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.146 - 0.989i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.471676 + 0.546727i\)
\(L(\frac12)\)  \(\approx\)  \(0.471676 + 0.546727i\)
\(L(2)\)   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 \( 1 + (1.95 + 2.27i)T \)
good5 \( 1 + (3.25 - 8.93i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (-0.410 + 2.32i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (-4.40 - 12.1i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (12.2 - 10.2i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (-12.2 + 7.04i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (3.29 - 5.70i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (25.9 - 4.58i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (-0.977 + 1.16i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (0.620 + 3.52i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (11.0 + 19.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-31.4 - 37.5i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (-78.8 + 28.6i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (34.9 + 6.16i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 - 65.8iT - 2.80e3T^{2} \)
59 \( 1 + (17.2 - 47.3i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (11.5 - 65.6i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (-72.5 + 60.8i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (-71.8 + 41.4i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (47.8 - 82.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (25.9 + 21.8i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (-7.14 + 8.50i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (8.83 + 5.10i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-66.7 + 24.2i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−14.06026757419974936841091164075, −12.32794095438381083727515703678, −11.76832526049485913580822554660, −10.73999068245297447158148574519, −9.829669313676575682914108060546, −7.54793464874844018365104938817, −7.26875177274501105806503418349, −6.13375582204157764238668958710, −4.21687112139819131911602392316, −2.34825679350537444399477677436, 0.57485318002860518000879171722, 3.80192954357316551459478186251, 5.00325928798118295211912756806, 5.86093257606209009729419720820, 7.982238435805699282845684107069, 8.868155538269986924645411485362, 9.879033282335512236476103373732, 11.25823849326988086730385116206, 12.19473724351329347223800071264, 12.73257259462957993613248469136

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