Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.866 - 0.5i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + 2.82i·5-s + 1.73i·7-s − 2.82i·8-s + (2.00 − 3.46i)10-s + 4.89·11-s − 13-s + (1.22 − 2.12i)14-s + (−2.00 + 3.46i)16-s + 2.82i·17-s − 5.19i·19-s + (−4.89 + 2.82i)20-s + (−5.99 − 3.46i)22-s − 4.89·23-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s + 1.26i·5-s + 0.654i·7-s − 0.999i·8-s + (0.632 − 1.09i)10-s + 1.47·11-s − 0.277·13-s + (0.327 − 0.566i)14-s + (−0.500 + 0.866i)16-s + 0.685i·17-s − 1.19i·19-s + (−1.09 + 0.632i)20-s + (−1.27 − 0.738i)22-s − 1.02·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.866 - 0.5i$
motivic weight  =  \(1\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 108,\ (\ :1/2),\ 0.866 - 0.5i)$
$L(1)$  $\approx$  $0.702105 + 0.188128i$
$L(\frac12)$  $\approx$  $0.702105 + 0.188128i$
$L(\frac{3}{2})$   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 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + 4.89T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 + 9.79T + 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−13.84615297786431326546519009863, −12.31226909877379339742711803629, −11.56634646519630827357679526066, −10.63309385102304075552945923961, −9.579514382579032216601020781947, −8.581188624764473361244357411680, −7.16132092439863928675358722446, −6.29241448203311121119305129276, −3.78108429902194902533382754795, −2.31677886908070896022890697102, 1.28738494146991810369837546020, 4.31239716399917921215721847652, 5.77710144178911363071619859598, 7.10773983113878985981094762058, 8.285813732700835825090870302135, 9.231739868664361363571273147442, 10.07056945536151663804908059052, 11.51030860027074266517156018103, 12.41742676422328489781301743072, 13.88899356930377682860245414798

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