Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.730 + 0.682i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.59 − 3.66i)2-s + (−10.9 − 11.6i)4-s + 29.4·5-s − 32.9i·7-s + (−60.2 + 21.4i)8-s + (46.9 − 108. i)10-s − 136. i·11-s + 164.·13-s + (−120. − 52.4i)14-s + (−17.2 + 255. i)16-s − 380.·17-s − 439. i·19-s + (−321. − 344. i)20-s + (−501. − 217. i)22-s − 171. i·23-s + ⋯
L(s)  = 1  + (0.398 − 0.917i)2-s + (−0.682 − 0.730i)4-s + 1.17·5-s − 0.671i·7-s + (−0.942 + 0.335i)8-s + (0.469 − 1.08i)10-s − 1.12i·11-s + 0.973·13-s + (−0.616 − 0.267i)14-s + (−0.0672 + 0.997i)16-s − 1.31·17-s − 1.21i·19-s + (−0.804 − 0.860i)20-s + (−1.03 − 0.449i)22-s − 0.324i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 + 0.682i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.730 + 0.682i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.730 + 0.682i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.797877 - 2.02175i\)
\(L(\frac12)\)  \(\approx\)  \(0.797877 - 2.02175i\)
\(L(3)\)   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 \( 1 + (-1.59 + 3.66i)T \)
3 \( 1 \)
good5 \( 1 - 29.4T + 625T^{2} \)
7 \( 1 + 32.9iT - 2.40e3T^{2} \)
11 \( 1 + 136. iT - 1.46e4T^{2} \)
13 \( 1 - 164.T + 2.85e4T^{2} \)
17 \( 1 + 380.T + 8.35e4T^{2} \)
19 \( 1 + 439. iT - 1.30e5T^{2} \)
23 \( 1 + 171. iT - 2.79e5T^{2} \)
29 \( 1 + 1.04e3T + 7.07e5T^{2} \)
31 \( 1 - 1.18e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.70e3T + 1.87e6T^{2} \)
41 \( 1 - 1.55e3T + 2.82e6T^{2} \)
43 \( 1 + 2.21e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.29e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.01e3T + 7.89e6T^{2} \)
59 \( 1 - 2.43e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.83e3T + 1.38e7T^{2} \)
67 \( 1 + 2.35e3iT - 2.01e7T^{2} \)
71 \( 1 + 884. iT - 2.54e7T^{2} \)
73 \( 1 - 6.92e3T + 2.83e7T^{2} \)
79 \( 1 - 1.03e4iT - 3.89e7T^{2} \)
83 \( 1 - 1.23e4iT - 4.74e7T^{2} \)
89 \( 1 + 5.75e3T + 6.27e7T^{2} \)
97 \( 1 + 8.15e3T + 8.85e7T^{2} \)
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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

−12.92024242549518275898270500899, −11.15190864873919601005453118566, −10.83675974900876786341596879056, −9.489061545548244781077366759718, −8.675897481523353949306975272493, −6.55334225842795847789218994608, −5.52636184760955872352193103158, −4.03225047245666487036544392363, −2.47933650845999230631295996225, −0.889730015685631032152157297536, 2.16461540014128999808426889621, 4.18320597105731006101292199956, 5.65907856891429800135980124351, 6.32535812662660819660401440930, 7.76471630056484019312501847506, 9.074832599135558663677964482904, 9.789925240637690500252582061951, 11.46368140526250258479993980618, 12.87384315608150736767331919748, 13.31023285002838406784901074150

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