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  =  \((2,\ 108,\ (\ :5/2),\ -0.998 - 0.0535i)\)
\(L(3)\)  \(\approx\)  \(0.0547249 + 2.04234i\)
\(L(\frac12)\)  \(\approx\)  \(0.0547249 + 2.04234i\)
\(L(\frac{7}{2})\)   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

−12.37149459385499632515058027428, −11.52041275988257437718015269879, −10.13901238592833214088367383503, −9.093660748840487362582447512855, −8.072329605058528355910437945531, −5.87847131245362095303048178113, −5.14353471307023965641075803614, −3.90141116352066118589973862766, −2.02801307237571778973182389555, −0.59423156121907396299609677975, 2.69782183354649139764567201834, 3.70716372565721186975821175974, 5.40050257848325696140993422018, 6.77211794090285629220614151313, 7.26011037017022059010497914810, 8.653233907994094890890514391484, 10.59274321119865413522701946935, 10.97067862780637003874060291590, 12.47166967399831356370728803415, 13.73415031158334167484719392072

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