Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.998 + 0.0535i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.10 − 3.89i)2-s + (1.71 + 31.9i)4-s − 98.6i·5-s − 67.7i·7-s + (117. − 137. i)8-s + (−383. + 404. i)10-s + 403.·11-s + 525.·13-s + (−263. + 278. i)14-s + (−1.01e3 + 109. i)16-s − 462. i·17-s − 1.95e3i·19-s + (3.15e3 − 168. i)20-s + (−1.65e3 − 1.56e3i)22-s − 696.·23-s + ⋯
L(s)  = 1  + (−0.725 − 0.687i)2-s + (0.0535 + 0.998i)4-s − 1.76i·5-s − 0.522i·7-s + (0.648 − 0.761i)8-s + (−1.21 + 1.28i)10-s + 1.00·11-s + 0.862·13-s + (−0.359 + 0.379i)14-s + (−0.994 + 0.106i)16-s − 0.388i·17-s − 1.24i·19-s + (1.76 − 0.0944i)20-s + (−0.729 − 0.691i)22-s − 0.274·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0535i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.998 + 0.0535i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.998 + 0.0535i)\)
\(L(3)\)  \(\approx\)  \(0.0290778 - 1.08519i\)
\(L(\frac12)\)  \(\approx\)  \(0.0290778 - 1.08519i\)
\(L(\frac{7}{2})\)   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 \( 1 + (4.10 + 3.89i)T \)
3 \( 1 \)
good5 \( 1 + 98.6iT - 3.12e3T^{2} \)
7 \( 1 + 67.7iT - 1.68e4T^{2} \)
11 \( 1 - 403.T + 1.61e5T^{2} \)
13 \( 1 - 525.T + 3.71e5T^{2} \)
17 \( 1 + 462. iT - 1.41e6T^{2} \)
19 \( 1 + 1.95e3iT - 2.47e6T^{2} \)
23 \( 1 + 696.T + 6.43e6T^{2} \)
29 \( 1 + 4.66e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.84e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.48e4T + 6.93e7T^{2} \)
41 \( 1 - 5.37e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.88e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.57e4T + 2.29e8T^{2} \)
53 \( 1 + 3.19e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.01e3T + 7.14e8T^{2} \)
61 \( 1 - 2.14e4T + 8.44e8T^{2} \)
67 \( 1 - 1.84e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.77e4T + 1.80e9T^{2} \)
73 \( 1 - 5.28e4T + 2.07e9T^{2} \)
79 \( 1 - 9.49e4iT - 3.07e9T^{2} \)
83 \( 1 - 8.22e4T + 3.93e9T^{2} \)
89 \( 1 + 2.06e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.98e4T + 8.58e9T^{2} \)
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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

−12.10523287409713599142576975930, −11.31424211480424334685513124847, −9.912778819116473805440618649562, −8.932985303091639461296489666435, −8.359216140600720681428615703199, −6.83533033976571671123521997169, −4.88293190740199366049307043525, −3.74113973081887250332461346194, −1.54070345266395264481326926896, −0.55798214124665453491109252392, 1.83686416694363202455111489179, 3.62005063379470164752914289363, 5.89674573153927217653684933092, 6.55564031710363591476313691966, 7.67637504098312476682683395571, 8.895552981689496554513232939200, 10.10077947197958248758599341748, 10.88847543293590406460625441298, 11.87588467873726031232207981415, 13.76988775079340383269345906126

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