Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.203 + 0.979i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.28 − 2.27i)2-s + (5.62 − 14.9i)4-s + (2.83 − 4.90i)5-s + (45.1 − 26.0i)7-s + (−15.6 − 62.0i)8-s + (−1.85 − 22.5i)10-s + (−92.3 + 53.3i)11-s + (61.0 − 105. i)13-s + (89.1 − 188. i)14-s + (−192. − 168. i)16-s + 122.·17-s − 593. i·19-s + (−57.5 − 69.9i)20-s + (−182. + 385. i)22-s + (−473. − 273. i)23-s + ⋯
L(s)  = 1  + (0.822 − 0.569i)2-s + (0.351 − 0.936i)4-s + (0.113 − 0.196i)5-s + (0.921 − 0.532i)7-s + (−0.244 − 0.969i)8-s + (−0.0185 − 0.225i)10-s + (−0.763 + 0.440i)11-s + (0.361 − 0.625i)13-s + (0.454 − 0.962i)14-s + (−0.752 − 0.658i)16-s + 0.424·17-s − 1.64i·19-s + (−0.143 − 0.174i)20-s + (−0.376 + 0.797i)22-s + (−0.895 − 0.516i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.203 + 0.979i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.203 + 0.979i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.81230 - 2.22724i\)
\(L(\frac12)\)  \(\approx\)  \(1.81230 - 2.22724i\)
\(L(3)\)   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 \( 1 + (-3.28 + 2.27i)T \)
3 \( 1 \)
good5 \( 1 + (-2.83 + 4.90i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-45.1 + 26.0i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (92.3 - 53.3i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-61.0 + 105. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 122.T + 8.35e4T^{2} \)
19 \( 1 + 593. iT - 1.30e5T^{2} \)
23 \( 1 + (473. + 273. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-367. - 637. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-507. - 292. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 2.28e3T + 1.87e6T^{2} \)
41 \( 1 + (1.43e3 - 2.48e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.94e3 - 1.12e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-913. + 527. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 4.75e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.86e3 - 1.07e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-33.1 - 57.4i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.55e3 - 2.05e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 5.03e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.70e3T + 2.83e7T^{2} \)
79 \( 1 + (1.19e3 - 690. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (2.60e3 - 1.50e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 3.18e3T + 6.27e7T^{2} \)
97 \( 1 + (-2.40e3 - 4.16e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−12.88605120756342843650219444386, −11.61826636573509188308108442470, −10.78543393478531162359703807190, −9.853951002004733642539439062077, −8.260802834064993179848433399835, −6.91484521233390817668162403000, −5.34037439808766619230846832024, −4.47391058970215508798693692644, −2.76434799592832301964163393603, −1.07551350475093829650154480012, 2.21953105141796474427891571533, 3.90701488819572958896201763966, 5.32141735366459352868087919219, 6.23842995496933458002777812391, 7.81542147173985090245349580586, 8.466646999074968312592981777295, 10.24912670789114055385239015438, 11.56530180053496917635797001120, 12.21576244865177264889664064800, 13.57434509103639328594339904898

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