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  =  \((2,\ 108,\ (\ :5/2),\ -0.617 - 0.786i)\)
\(L(3)\)  \(\approx\)  \(1.20273 + 2.47521i\)
\(L(\frac12)\)  \(\approx\)  \(1.20273 + 2.47521i\)
\(L(\frac{7}{2})\)   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

−13.15554989865250718061266747613, −12.27010701526772574537886626514, −11.27430364597327623855518402590, −10.12763931493199155687859852167, −8.480298817807397028862665546559, −7.39118298920124748996844463515, −6.24241757523849385751608092396, −5.14751353518687443510488655302, −3.61399850786706201935954151060, −2.30197582416654672140997743798, 0.76534862495844142980071814979, 2.49158167117141460440320834680, 4.11981153968802261214097459214, 5.13976874830713937966445837650, 6.47154267383303365433182425097, 7.82651011623443371233178958421, 9.401257161864270359115926568747, 10.46929473321889851737089292433, 11.49227025502148812456644828966, 12.53612140265771806866991933379

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