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  :  $\chi_{108} (91, \cdot )$
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

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

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