Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.715 - 0.698i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.277 − 1.98i)2-s + (−2.64 − 1.42i)3-s + (−3.84 − 1.09i)4-s + (−0.542 − 3.07i)5-s + (−3.55 + 4.83i)6-s + (−2.98 + 3.56i)7-s + (−3.24 + 7.31i)8-s + (4.94 + 7.51i)9-s + (−6.24 + 0.220i)10-s + (2.54 + 0.448i)11-s + (8.59 + 8.37i)12-s + (−16.7 − 6.09i)13-s + (6.22 + 6.90i)14-s + (−2.94 + 8.89i)15-s + (13.5 + 8.45i)16-s + (−10.3 − 17.9i)17-s + ⋯
L(s)  = 1  + (0.138 − 0.990i)2-s + (−0.880 − 0.474i)3-s + (−0.961 − 0.274i)4-s + (−0.108 − 0.615i)5-s + (−0.592 + 0.805i)6-s + (−0.426 + 0.508i)7-s + (−0.405 + 0.914i)8-s + (0.549 + 0.835i)9-s + (−0.624 + 0.0220i)10-s + (0.231 + 0.0408i)11-s + (0.715 + 0.698i)12-s + (−1.28 − 0.468i)13-s + (0.444 + 0.493i)14-s + (−0.196 + 0.592i)15-s + (0.848 + 0.528i)16-s + (−0.609 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.715 - 0.698i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.715 - 0.698i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.159530 + 0.392047i\)
\(L(\frac12)\)  \(\approx\)  \(0.159530 + 0.392047i\)
\(L(2)\)   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 \( 1 + (-0.277 + 1.98i)T \)
3 \( 1 + (2.64 + 1.42i)T \)
good5 \( 1 + (0.542 + 3.07i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (2.98 - 3.56i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (-2.54 - 0.448i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (16.7 + 6.09i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (10.3 + 17.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (24.5 + 14.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-14.6 - 17.4i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-5.90 + 2.14i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (22.3 + 26.6i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (22.9 + 39.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (4.27 + 1.55i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (12.6 + 2.23i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-1.37 + 1.64i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 31.8T + 2.80e3T^{2} \)
59 \( 1 + (-111. + 19.6i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-63.3 - 53.1i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (-10.8 + 29.8i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (0.0648 - 0.0374i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (60.1 - 104. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (0.117 + 0.323i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (32.5 + 89.3i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (-59.4 + 102. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-11.2 + 63.5i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−12.72489357664392322966698697991, −11.86118149268767849192499522427, −10.98680752663204071019685467578, −9.754610876076640011493653314039, −8.748656864546878763524105488120, −7.10380500269403468230898519048, −5.50309674713277386825297616283, −4.58014100876519937472505519214, −2.39818189267944063419009908780, −0.32315566498902076177870235657, 3.80213756955187013705589037608, 4.94305626796337063751461149720, 6.49965047703463905015028236556, 6.93976576061417354388761437017, 8.637263125781377888000244162234, 9.964035562945888826430740124558, 10.72857686638980581954762573274, 12.23363683910630498614574861538, 13.04106598686919119495437881703, 14.64236747293037623248277065982

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