Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.130 + 0.991i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.64 − 0.370i)2-s + (31.7 − 4.17i)4-s − 38.9i·5-s − 132. i·7-s + (177. − 35.3i)8-s + (−14.4 − 219. i)10-s − 190.·11-s − 416.·13-s + (−49.0 − 747. i)14-s + (989. − 265. i)16-s − 1.14e3i·17-s − 768. i·19-s + (−162. − 1.23e3i)20-s + (−1.07e3 + 70.4i)22-s + 2.23e3·23-s + ⋯
L(s)  = 1  + (0.997 − 0.0654i)2-s + (0.991 − 0.130i)4-s − 0.696i·5-s − 1.02i·7-s + (0.980 − 0.195i)8-s + (−0.0455 − 0.694i)10-s − 0.474·11-s − 0.683·13-s + (−0.0668 − 1.01i)14-s + (0.965 − 0.258i)16-s − 0.961i·17-s − 0.488i·19-s + (−0.0908 − 0.690i)20-s + (−0.473 + 0.0310i)22-s + 0.879·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.130 + 0.991i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.130 + 0.991i)\)
\(L(3)\)  \(\approx\)  \(2.47607 - 2.17137i\)
\(L(\frac12)\)  \(\approx\)  \(2.47607 - 2.17137i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-5.64 + 0.370i)T \)
3 \( 1 \)
good5 \( 1 + 38.9iT - 3.12e3T^{2} \)
7 \( 1 + 132. iT - 1.68e4T^{2} \)
11 \( 1 + 190.T + 1.61e5T^{2} \)
13 \( 1 + 416.T + 3.71e5T^{2} \)
17 \( 1 + 1.14e3iT - 1.41e6T^{2} \)
19 \( 1 + 768. iT - 2.47e6T^{2} \)
23 \( 1 - 2.23e3T + 6.43e6T^{2} \)
29 \( 1 - 5.91e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.00e4iT - 2.86e7T^{2} \)
37 \( 1 - 4.31e3T + 6.93e7T^{2} \)
41 \( 1 - 1.85e4iT - 1.15e8T^{2} \)
43 \( 1 + 5.56e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.26e4T + 2.29e8T^{2} \)
53 \( 1 - 3.09e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.92e4T + 7.14e8T^{2} \)
61 \( 1 + 4.52e4T + 8.44e8T^{2} \)
67 \( 1 - 2.47e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.87e4T + 1.80e9T^{2} \)
73 \( 1 - 8.12e4T + 2.07e9T^{2} \)
79 \( 1 - 3.33e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.75e4T + 3.93e9T^{2} \)
89 \( 1 - 6.54e4iT - 5.58e9T^{2} \)
97 \( 1 + 2.91e4T + 8.58e9T^{2} \)
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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

−12.82564276456814909979127271760, −11.60596473063036152916066351174, −10.66447271256224357285871785247, −9.450787588572756453633304615544, −7.75431542118885451084876924170, −6.84630747992690558780343535381, −5.22070686688102887014456974457, −4.40414189351339367118478026531, −2.81087066518356807962254115012, −0.919836542125674704351169530544, 2.15625433743368561657857688147, 3.28562514592938517087796356123, 4.95368381725509063772323844882, 6.04460674469842915834556219238, 7.15547260037529757341789633911, 8.451005939681268209832626189599, 10.12356573152309078543052148836, 11.08308968058610314702302881497, 12.18643388159394798743346723539, 12.88057814725918807985636309805

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