Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.774 + 0.632i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.84 − 0.778i)2-s + (2.78 − 2.86i)4-s + (−1.10 + 1.90i)5-s + (7.23 − 4.17i)7-s + (2.90 − 7.45i)8-s + (−0.544 + 4.36i)10-s + (4.54 − 2.62i)11-s + (−7.37 + 12.7i)13-s + (10.0 − 13.3i)14-s + (−0.450 − 15.9i)16-s − 28.2·17-s + 19.1i·19-s + (2.39 + 8.47i)20-s + (6.33 − 8.37i)22-s + (−3.16 − 1.82i)23-s + ⋯
L(s)  = 1  + (0.921 − 0.389i)2-s + (0.697 − 0.716i)4-s + (−0.220 + 0.381i)5-s + (1.03 − 0.597i)7-s + (0.363 − 0.931i)8-s + (−0.0544 + 0.436i)10-s + (0.413 − 0.238i)11-s + (−0.567 + 0.982i)13-s + (0.720 − 0.952i)14-s + (−0.0281 − 0.999i)16-s − 1.66·17-s + 1.00i·19-s + (0.119 + 0.423i)20-s + (0.287 − 0.380i)22-s + (−0.137 − 0.0794i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.774 + 0.632i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.774 + 0.632i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.14986 - 0.765714i\)
\(L(\frac12)\)  \(\approx\)  \(2.14986 - 0.765714i\)
\(L(2)\)   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 \( 1 + (-1.84 + 0.778i)T \)
3 \( 1 \)
good5 \( 1 + (1.10 - 1.90i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-7.23 + 4.17i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-4.54 + 2.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.37 - 12.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 28.2T + 289T^{2} \)
19 \( 1 - 19.1iT - 361T^{2} \)
23 \( 1 + (3.16 + 1.82i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-12.3 - 21.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (32.9 + 19.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 4.21T + 1.36e3T^{2} \)
41 \( 1 + (-9.92 + 17.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.1 + 11.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.8 - 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 32.1T + 2.80e3T^{2} \)
59 \( 1 + (-7.96 - 4.59i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (40.8 + 70.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.86 - 3.96i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 62.9iT - 5.04e3T^{2} \)
73 \( 1 - 33.3T + 5.32e3T^{2} \)
79 \( 1 + (-53.7 + 31.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-103. + 59.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 107.T + 7.92e3T^{2} \)
97 \( 1 + (-1.78 - 3.09i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.49786089941613009940647349474, −12.23701520284924939760855732897, −11.25238010116248575423535023202, −10.70380400475800851327720926592, −9.177312134236502288321837576840, −7.50615865741608720606512709434, −6.49566095960801606467864287849, −4.87176358146779922439016780308, −3.84709874097132258207358421828, −1.90088102603402957475671973919, 2.41955482269049163010347208014, 4.40027408612530290173247774511, 5.25485989631086839116026862832, 6.72610922315317839455316533705, 7.986600792637740619277805022258, 8.923680315405402432158330954324, 10.82547171837280879280131518741, 11.72343128070076313602980222235, 12.60671573938233310630384422344, 13.56531073068516567490147160180

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