Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.0834 + 0.996i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.82 + 4.16i)2-s + (−2.67 + 31.8i)4-s − 46.0i·5-s − 134. i·7-s + (−142. + 110. i)8-s + (191. − 176. i)10-s − 471.·11-s − 987.·13-s + (561. − 516. i)14-s + (−1.00e3 − 170. i)16-s − 1.46e3i·17-s − 308. i·19-s + (1.46e3 + 122. i)20-s + (−1.80e3 − 1.96e3i)22-s + 2.11e3·23-s + ⋯
L(s)  = 1  + (0.676 + 0.736i)2-s + (−0.0834 + 0.996i)4-s − 0.823i·5-s − 1.03i·7-s + (−0.789 + 0.613i)8-s + (0.606 − 0.557i)10-s − 1.17·11-s − 1.61·13-s + (0.765 − 0.703i)14-s + (−0.986 − 0.166i)16-s − 1.23i·17-s − 0.195i·19-s + (0.820 + 0.0687i)20-s + (−0.795 − 0.864i)22-s + 0.835·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0834 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0834 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.0834 + 0.996i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.0834 + 0.996i)\)
\(L(3)\)  \(\approx\)  \(0.697171 - 0.757998i\)
\(L(\frac12)\)  \(\approx\)  \(0.697171 - 0.757998i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-3.82 - 4.16i)T \)
3 \( 1 \)
good5 \( 1 + 46.0iT - 3.12e3T^{2} \)
7 \( 1 + 134. iT - 1.68e4T^{2} \)
11 \( 1 + 471.T + 1.61e5T^{2} \)
13 \( 1 + 987.T + 3.71e5T^{2} \)
17 \( 1 + 1.46e3iT - 1.41e6T^{2} \)
19 \( 1 + 308. iT - 2.47e6T^{2} \)
23 \( 1 - 2.11e3T + 6.43e6T^{2} \)
29 \( 1 + 5.19e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.36e3iT - 2.86e7T^{2} \)
37 \( 1 + 8.74e3T + 6.93e7T^{2} \)
41 \( 1 + 1.39e3iT - 1.15e8T^{2} \)
43 \( 1 - 9.41e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.11e4T + 2.29e8T^{2} \)
53 \( 1 + 2.84e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.50e4T + 7.14e8T^{2} \)
61 \( 1 + 3.78e4T + 8.44e8T^{2} \)
67 \( 1 - 6.52e4iT - 1.35e9T^{2} \)
71 \( 1 + 9.86e3T + 1.80e9T^{2} \)
73 \( 1 + 5.48e4T + 2.07e9T^{2} \)
79 \( 1 + 9.16e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.51e4T + 3.93e9T^{2} \)
89 \( 1 - 7.31e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.14e4T + 8.58e9T^{2} \)
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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

−12.77250658915138959952701974603, −11.79744793547146522743639115043, −10.31872646396563291465685463766, −9.028064001630348004695859897167, −7.69707188681913194449319748970, −7.03807757609937061103469579775, −5.18653342221317254977090637424, −4.63069790480767852268816297951, −2.83680524468104926567799548072, −0.28724925895830211573003747896, 2.18032294409177030409578709982, 3.07859661330563312831905005029, 4.89090831238539443443206822812, 5.90863034443410370126994881179, 7.32959359775913395643615346199, 8.972311410287915967720441603360, 10.23369875503212889592135870022, 10.88858957766006844749719879007, 12.21892252750168544396508195736, 12.73421044782946678850555095064

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