Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.493 - 0.869i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0678 + 3.99i)2-s + (−15.9 − 0.542i)4-s + (−16.6 − 28.7i)5-s + (39.9 + 23.0i)7-s + (3.25 − 63.9i)8-s + (116. − 64.4i)10-s + (63.6 + 36.7i)11-s + (151. + 262. i)13-s + (−95.0 + 158. i)14-s + (255. + 17.3i)16-s + 182.·17-s − 314. i·19-s + (250. + 469. i)20-s + (−151. + 251. i)22-s + (290. − 167. i)23-s + ⋯
L(s)  = 1  + (−0.0169 + 0.999i)2-s + (−0.999 − 0.0339i)4-s + (−0.664 − 1.15i)5-s + (0.815 + 0.471i)7-s + (0.0508 − 0.998i)8-s + (1.16 − 0.644i)10-s + (0.525 + 0.303i)11-s + (0.896 + 1.55i)13-s + (−0.484 + 0.807i)14-s + (0.997 + 0.0677i)16-s + 0.629·17-s − 0.870i·19-s + (0.625 + 1.17i)20-s + (−0.312 + 0.520i)22-s + (0.549 − 0.317i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.493 - 0.869i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.493 - 0.869i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.36324 + 0.793433i\)
\(L(\frac12)\)  \(\approx\)  \(1.36324 + 0.793433i\)
\(L(3)\)   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 \( 1 + (0.0678 - 3.99i)T \)
3 \( 1 \)
good5 \( 1 + (16.6 + 28.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-39.9 - 23.0i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-63.6 - 36.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-151. - 262. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 182.T + 8.35e4T^{2} \)
19 \( 1 + 314. iT - 1.30e5T^{2} \)
23 \( 1 + (-290. + 167. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (357. - 618. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-985. + 568. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.00e3T + 1.87e6T^{2} \)
41 \( 1 + (-557. - 965. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-2.18e3 - 1.25e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (980. + 566. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 1.05e3T + 7.89e6T^{2} \)
59 \( 1 + (878. - 507. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (430. - 745. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-559. + 322. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 9.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.89e3T + 2.83e7T^{2} \)
79 \( 1 + (6.76e3 + 3.90e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-7.05e3 - 4.07e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 7.65e3T + 6.27e7T^{2} \)
97 \( 1 + (6.36e3 - 1.10e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.29884387952047465020500359318, −12.21107385185110993551703938952, −11.31344453100092066092574309663, −9.316685714942198721442394647309, −8.746114246434679158721417549033, −7.76447617346977358412945666101, −6.41596090557727148510567079805, −4.94247330685314476725040561906, −4.18311904255276478535197630988, −1.13409779500929128504192943748, 1.02028424773004737401682230486, 3.04589786265483846174097865661, 4.00257009168146121355745627703, 5.74036479620076693075357074052, 7.57714222541508393558869602371, 8.378350938481854054796116560817, 10.04733039719181573096635941046, 10.88614834248582740830076984457, 11.44921063763272024520946545544, 12.62815963764482562489211041133

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