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  =  $(2,\ 108,\ (\ :1/2),\ 0.866 - 0.5i)$
$L(1)$  $\approx$  $1.46275 + 0.391945i$
$L(\frac12)$  $\approx$  $1.46275 + 0.391945i$
$L(\frac{3}{2})$   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

−13.53462085649286161955794118302, −12.89531606599519667157622523628, −12.14263861958532953022595243991, −10.93906277710774321682413148161, −9.163159153657917161863809723513, −8.281721265933742721789710956170, −7.03252381209453066427768669185, −5.28584919165466854143316822829, −4.90535018477208437957651904602, −2.77828702923805117006288224857, 2.53358746501207533779415162693, 3.83954399237016636898470666470, 5.46966003214090417135767742386, 6.73714895808816940727636904501, 7.80511589406951707490348315572, 10.03972817666480461842307116961, 10.52750906299309664461428638242, 11.42675846969452617444052018382, 12.77112879581487018740784099000, 13.56591702545586773458051875234

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