Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.183 + 0.983i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.701 + 3.93i)2-s + (−15.0 + 5.52i)4-s + (−14.3 + 24.7i)5-s + (−22.2 + 12.8i)7-s + (−32.3 − 55.2i)8-s + (−107. − 38.9i)10-s + (93.9 − 54.2i)11-s + (44.2 − 76.5i)13-s + (−66.0 − 78.4i)14-s + (194. − 165. i)16-s − 504.·17-s − 191. i·19-s + (77.8 − 451. i)20-s + (279. + 331. i)22-s + (−831. − 480. i)23-s + ⋯
L(s)  = 1  + (0.175 + 0.984i)2-s + (−0.938 + 0.345i)4-s + (−0.572 + 0.991i)5-s + (−0.453 + 0.261i)7-s + (−0.504 − 0.863i)8-s + (−1.07 − 0.389i)10-s + (0.776 − 0.448i)11-s + (0.261 − 0.453i)13-s + (−0.337 − 0.400i)14-s + (0.761 − 0.648i)16-s − 1.74·17-s − 0.530i·19-s + (0.194 − 1.12i)20-s + (0.577 + 0.685i)22-s + (−1.57 − 0.907i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.183 + 0.983i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.183 + 0.983i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.139790 - 0.168314i\)
\(L(\frac12)\)  \(\approx\)  \(0.139790 - 0.168314i\)
\(L(3)\)   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 \( 1 + (-0.701 - 3.93i)T \)
3 \( 1 \)
good5 \( 1 + (14.3 - 24.7i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (22.2 - 12.8i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-93.9 + 54.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-44.2 + 76.5i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 504.T + 8.35e4T^{2} \)
19 \( 1 + 191. iT - 1.30e5T^{2} \)
23 \( 1 + (831. + 480. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-396. - 687. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-285. - 164. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 209.T + 1.87e6T^{2} \)
41 \( 1 + (-528. + 914. i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (2.88e3 - 1.66e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (977. - 564. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.13e3T + 7.89e6T^{2} \)
59 \( 1 + (4.03e3 + 2.33e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.79e3 - 4.84e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (6.12e3 + 3.53e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 4.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.95e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.52e3 + 879. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-2.62e3 + 1.51e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 559.T + 6.27e7T^{2} \)
97 \( 1 + (-1.10e3 - 1.90e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−13.96598820414350051289917277309, −12.93584677994051867063590338702, −11.71850683147113188448389967986, −10.56032893835853606572787909866, −9.151876430532316788380858343077, −8.168808482905231260156751077982, −6.80579846806749033467924362323, −6.23462757260635748677975838863, −4.41327574590679884069335417924, −3.12592385104325366636172917194, 0.091755453143157531526890363833, 1.77013396024763557364279519874, 3.84214522786289751743099056486, 4.61443298516899598425969773394, 6.34509201166847510816964558119, 8.177133908600341714047542349121, 9.147840568788194743042155956944, 10.07520817952019373999098888474, 11.51893816014971735831538420611, 12.07050800801645281411858168959

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