Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.706 - 0.707i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.93 + 0.717i)2-s + (14.9 + 5.64i)4-s + (5.51 + 9.55i)5-s + (−10.3 − 5.95i)7-s + (54.8 + 32.9i)8-s + (14.8 + 41.5i)10-s + (189. + 109. i)11-s + (18.5 + 32.1i)13-s + (−36.3 − 30.8i)14-s + (192. + 169. i)16-s − 284.·17-s − 45.4i·19-s + (28.6 + 174. i)20-s + (668. + 567. i)22-s + (174. − 100. i)23-s + ⋯
L(s)  = 1  + (0.983 + 0.179i)2-s + (0.935 + 0.352i)4-s + (0.220 + 0.382i)5-s + (−0.210 − 0.121i)7-s + (0.857 + 0.514i)8-s + (0.148 + 0.415i)10-s + (1.57 + 0.906i)11-s + (0.109 + 0.189i)13-s + (−0.185 − 0.157i)14-s + (0.751 + 0.660i)16-s − 0.982·17-s − 0.126i·19-s + (0.0716 + 0.435i)20-s + (1.38 + 1.17i)22-s + (0.329 − 0.190i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.706 - 0.707i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.706 - 0.707i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(3.16754 + 1.31388i\)
\(L(\frac12)\)  \(\approx\)  \(3.16754 + 1.31388i\)
\(L(3)\)   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 \( 1 + (-3.93 - 0.717i)T \)
3 \( 1 \)
good5 \( 1 + (-5.51 - 9.55i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (10.3 + 5.95i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-189. - 109. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-18.5 - 32.1i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 284.T + 8.35e4T^{2} \)
19 \( 1 + 45.4iT - 1.30e5T^{2} \)
23 \( 1 + (-174. + 100. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (614. - 1.06e3i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-1.31e3 + 757. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.52e3T + 1.87e6T^{2} \)
41 \( 1 + (1.31e3 + 2.28e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-34.6 - 19.9i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (2.49e3 + 1.44e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.41e3T + 7.89e6T^{2} \)
59 \( 1 + (-2.45e3 + 1.41e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.62e3 + 4.55e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (805. - 465. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 1.16e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.16e3T + 2.83e7T^{2} \)
79 \( 1 + (6.48e3 + 3.74e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (966. + 558. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 6.73e3T + 6.27e7T^{2} \)
97 \( 1 + (6.02e3 - 1.04e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.20233858789619067607308066255, −12.17816318310001575482625188078, −11.30550947996464838480935971694, −10.10429289685641500611361707877, −8.728182601775534937700250901842, −6.96094972811652265180098758403, −6.51604070902916826241755631566, −4.82661370010730626968508717475, −3.64093276527389697582394714111, −1.95523525211804084622868836637, 1.35836701197522479458557161916, 3.21895149098569566167770195870, 4.52088174372875620881092052130, 5.93176385498272714993031293593, 6.80759456419666505059380095841, 8.555649767208266211302889962064, 9.731211818978908122375811634201, 11.15739221893937583965973426629, 11.82938329101551121495274965146, 13.02383321260894225926576709140

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