Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.617 - 0.786i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.08 + 2.47i)2-s + (19.7 − 25.1i)4-s + 33.1i·5-s − 59.4i·7-s + (−38.4 + 176. i)8-s + (−81.9 − 168. i)10-s + 179.·11-s − 143.·13-s + (146. + 302. i)14-s + (−241. − 995. i)16-s − 93.8i·17-s − 1.61e3i·19-s + (834. + 655. i)20-s + (−911. + 442. i)22-s + 2.41e3·23-s + ⋯
L(s)  = 1  + (−0.899 + 0.437i)2-s + (0.617 − 0.786i)4-s + 0.593i·5-s − 0.458i·7-s + (−0.212 + 0.977i)8-s + (−0.259 − 0.533i)10-s + 0.446·11-s − 0.235·13-s + (0.200 + 0.412i)14-s + (−0.236 − 0.971i)16-s − 0.0787i·17-s − 1.02i·19-s + (0.466 + 0.366i)20-s + (−0.401 + 0.195i)22-s + 0.951·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.617 - 0.786i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.617 - 0.786i)\)
\(L(3)\)  \(\approx\)  \(1.06936 + 0.519614i\)
\(L(\frac12)\)  \(\approx\)  \(1.06936 + 0.519614i\)
\(L(\frac{7}{2})\)   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 \( 1 + (5.08 - 2.47i)T \)
3 \( 1 \)
good5 \( 1 - 33.1iT - 3.12e3T^{2} \)
7 \( 1 + 59.4iT - 1.68e4T^{2} \)
11 \( 1 - 179.T + 1.61e5T^{2} \)
13 \( 1 + 143.T + 3.71e5T^{2} \)
17 \( 1 + 93.8iT - 1.41e6T^{2} \)
19 \( 1 + 1.61e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.41e3T + 6.43e6T^{2} \)
29 \( 1 - 7.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.72e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.38e3T + 6.93e7T^{2} \)
41 \( 1 - 1.18e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.95e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.37e3T + 2.29e8T^{2} \)
53 \( 1 + 2.68e4iT - 4.18e8T^{2} \)
59 \( 1 - 6.14e3T + 7.14e8T^{2} \)
61 \( 1 - 5.52e4T + 8.44e8T^{2} \)
67 \( 1 + 3.88e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.52e4T + 1.80e9T^{2} \)
73 \( 1 - 2.10e4T + 2.07e9T^{2} \)
79 \( 1 - 7.64e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.85e4T + 3.93e9T^{2} \)
89 \( 1 + 1.85e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.37e5T + 8.58e9T^{2} \)
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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

−12.94946728524265174707178242723, −11.40076340670329054510947531406, −10.73475051443707785447393502991, −9.612949430915241164807867295702, −8.607037036551296909625732987518, −7.22177723289156436813602436500, −6.60896755354801764842954607258, −4.97096073065524571441525808144, −2.89337951459957598895076648727, −1.04263375538950741189301661293, 0.795006178458790688793342862122, 2.34409145973516800983621999572, 4.04819942227570358646266433904, 5.85862072058812399938109653733, 7.32815459830175702438061685179, 8.494476381345074459046841281371, 9.303715382468068232402810006504, 10.34990079445942988882358458371, 11.60339887791023764987824562745, 12.32352778086235626947088482840

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