Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.981 + 0.189i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.54 − 3.08i)2-s + (−3.02 − 15.7i)4-s − 3.58·5-s − 16.2i·7-s + (−56.1 − 30.6i)8-s + (−9.12 + 11.0i)10-s − 144. i·11-s − 223.·13-s + (−50.1 − 41.4i)14-s + (−237. + 95.0i)16-s + 1.80·17-s + 70.1i·19-s + (10.8 + 56.2i)20-s + (−446. − 368. i)22-s − 251. i·23-s + ⋯
L(s)  = 1  + (0.636 − 0.771i)2-s + (−0.189 − 0.981i)4-s − 0.143·5-s − 0.331i·7-s + (−0.877 − 0.479i)8-s + (−0.0912 + 0.110i)10-s − 1.19i·11-s − 1.32·13-s + (−0.255 − 0.211i)14-s + (−0.928 + 0.371i)16-s + 0.00624·17-s + 0.194i·19-s + (0.0271 + 0.140i)20-s + (−0.922 − 0.761i)22-s − 0.475i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.981 + 0.189i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.981 + 0.189i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.150698 - 1.57926i\)
\(L(\frac12)\)  \(\approx\)  \(0.150698 - 1.57926i\)
\(L(3)\)   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 \( 1 + (-2.54 + 3.08i)T \)
3 \( 1 \)
good5 \( 1 + 3.58T + 625T^{2} \)
7 \( 1 + 16.2iT - 2.40e3T^{2} \)
11 \( 1 + 144. iT - 1.46e4T^{2} \)
13 \( 1 + 223.T + 2.85e4T^{2} \)
17 \( 1 - 1.80T + 8.35e4T^{2} \)
19 \( 1 - 70.1iT - 1.30e5T^{2} \)
23 \( 1 + 251. iT - 2.79e5T^{2} \)
29 \( 1 - 1.64e3T + 7.07e5T^{2} \)
31 \( 1 + 1.04e3iT - 9.23e5T^{2} \)
37 \( 1 + 690.T + 1.87e6T^{2} \)
41 \( 1 - 993.T + 2.82e6T^{2} \)
43 \( 1 - 1.98e3iT - 3.41e6T^{2} \)
47 \( 1 + 3.19e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.86e3T + 7.89e6T^{2} \)
59 \( 1 + 5.57e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.83e3T + 1.38e7T^{2} \)
67 \( 1 + 5.28e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.87e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.52e3T + 2.83e7T^{2} \)
79 \( 1 - 7.36e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.00e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.14e4T + 6.27e7T^{2} \)
97 \( 1 + 9.02e3T + 8.85e7T^{2} \)
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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

−12.39180195308435266218562655764, −11.58446042848754879645030829827, −10.51915333516767260798787752755, −9.599122016196554013458475359083, −8.176405755352219224524737590736, −6.57066490376352129148187504355, −5.26942024810122226205747411111, −3.95850747820891187487413393757, −2.53824696834225361095194473345, −0.55839093073197238344604256784, 2.56509818798426279678375315515, 4.32368498766556182908580950079, 5.35058221870560297096544513968, 6.83587000327107104630195341284, 7.67341654831033221993617807460, 8.988511413976721782503146315777, 10.17525548323902412143282766949, 12.00360392142518081720234614568, 12.31639698929965631732934276442, 13.62701106987865162289146809035

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