Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.236 + 0.971i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.63 + 3.01i)2-s + (−2.15 − 15.8i)4-s + (10.5 + 18.3i)5-s + (−38.6 − 22.3i)7-s + (53.4 + 35.2i)8-s + (−83.0 − 16.3i)10-s + (−58.6 − 33.8i)11-s + (14.5 + 25.2i)13-s + (168. − 57.7i)14-s + (−246. + 68.2i)16-s − 402.·17-s − 644. i·19-s + (267. − 207. i)20-s + (256. − 87.6i)22-s + (335. − 193. i)23-s + ⋯
L(s)  = 1  + (−0.657 + 0.753i)2-s + (−0.134 − 0.990i)4-s + (0.423 + 0.732i)5-s + (−0.788 − 0.455i)7-s + (0.834 + 0.550i)8-s + (−0.830 − 0.163i)10-s + (−0.485 − 0.280i)11-s + (0.0861 + 0.149i)13-s + (0.861 − 0.294i)14-s + (−0.963 + 0.266i)16-s − 1.39·17-s − 1.78i·19-s + (0.669 − 0.517i)20-s + (0.529 − 0.181i)22-s + (0.634 − 0.366i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.236 + 0.971i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.236 + 0.971i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.431247 - 0.339003i\)
\(L(\frac12)\)  \(\approx\)  \(0.431247 - 0.339003i\)
\(L(3)\)   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 \( 1 + (2.63 - 3.01i)T \)
3 \( 1 \)
good5 \( 1 + (-10.5 - 18.3i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (38.6 + 22.3i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (58.6 + 33.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-14.5 - 25.2i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 402.T + 8.35e4T^{2} \)
19 \( 1 + 644. iT - 1.30e5T^{2} \)
23 \( 1 + (-335. + 193. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-362. + 627. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-1.09e3 + 629. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.40e3T + 1.87e6T^{2} \)
41 \( 1 + (-774. - 1.34e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.62e3 + 935. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (3.61e3 + 2.08e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 906.T + 7.89e6T^{2} \)
59 \( 1 + (3.91e3 - 2.26e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.31e3 - 2.27e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-58.7 + 33.8i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 1.31e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.47e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.78e3 - 2.18e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (659. + 381. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 8.08e3T + 6.27e7T^{2} \)
97 \( 1 + (3.33e3 - 5.77e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.31454954725271418730567678690, −11.27323352756702432264051510302, −10.45298036142790849871374957995, −9.506535688770986571466972270131, −8.402459404495901369563120380335, −6.85137259190925567663810457296, −6.47141109685061800830409896817, −4.76155089466294483854300116816, −2.60454195574296345737469555816, −0.29759844352536684800643871983, 1.62304414920337446222643357633, 3.19848789403749150745008317272, 4.90626488896197444563689954494, 6.55254674662548444826902163063, 8.119520878068228730691229849166, 9.075661719170774502604674691006, 9.909850586181147092623186645688, 10.96096158283121781928400910984, 12.37596350731814536672252450227, 12.78938987812990063077556601056

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