Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.574 + 0.818i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.81 − 1.20i)2-s + (13.0 − 9.19i)4-s + 13.3·5-s − 25.4i·7-s + (38.8 − 50.8i)8-s + (51.0 − 16.1i)10-s − 34.7i·11-s + 116.·13-s + (−30.6 − 96.9i)14-s + (86.9 − 240. i)16-s + 109.·17-s + 264. i·19-s + (175. − 123. i)20-s + (−41.8 − 132. i)22-s − 890. i·23-s + ⋯
L(s)  = 1  + (0.953 − 0.301i)2-s + (0.818 − 0.574i)4-s + 0.535·5-s − 0.518i·7-s + (0.607 − 0.794i)8-s + (0.510 − 0.161i)10-s − 0.287i·11-s + 0.691·13-s + (−0.156 − 0.494i)14-s + (0.339 − 0.940i)16-s + 0.379·17-s + 0.733i·19-s + (0.438 − 0.307i)20-s + (−0.0865 − 0.273i)22-s − 1.68i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.574 + 0.818i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.574 + 0.818i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(3.04284 - 1.58151i\)
\(L(\frac12)\)  \(\approx\)  \(3.04284 - 1.58151i\)
\(L(3)\)   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 \( 1 + (-3.81 + 1.20i)T \)
3 \( 1 \)
good5 \( 1 - 13.3T + 625T^{2} \)
7 \( 1 + 25.4iT - 2.40e3T^{2} \)
11 \( 1 + 34.7iT - 1.46e4T^{2} \)
13 \( 1 - 116.T + 2.85e4T^{2} \)
17 \( 1 - 109.T + 8.35e4T^{2} \)
19 \( 1 - 264. iT - 1.30e5T^{2} \)
23 \( 1 + 890. iT - 2.79e5T^{2} \)
29 \( 1 - 1.11e3T + 7.07e5T^{2} \)
31 \( 1 - 1.66e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.98e3T + 1.87e6T^{2} \)
41 \( 1 + 1.22e3T + 2.82e6T^{2} \)
43 \( 1 - 3.14e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.36e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.18e3T + 7.89e6T^{2} \)
59 \( 1 - 1.80e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.51e3T + 1.38e7T^{2} \)
67 \( 1 - 5.25e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.12e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.35e3T + 2.83e7T^{2} \)
79 \( 1 + 445. iT - 3.89e7T^{2} \)
83 \( 1 + 1.11e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.03e4T + 6.27e7T^{2} \)
97 \( 1 - 474.T + 8.85e7T^{2} \)
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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

−12.88330847576978617794737788562, −11.98864395200690894221227827850, −10.69817816982868631470539634028, −10.08498602924773609272516941312, −8.413836684920517924186750475831, −6.82804929183674780347340587990, −5.85878732462221741495620559580, −4.49202183686764392193624644250, −3.09638534976922843669367291848, −1.35206977739720704030771945884, 2.01806744814795316124852776432, 3.58627706672875925549317178601, 5.17483530317422743313876580482, 6.09565106074367588502117056568, 7.34698437225824385746148495963, 8.674958063762506938485037695232, 10.04411508459867585582352562168, 11.39101155160983779761158937616, 12.19732326065993010070597117277, 13.43757805858741229647567091434

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