Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
Sign $0.820 + 0.570i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 − 0.863i)2-s + (−1.22 + 1.22i)3-s + (2.50 + 3.11i)4-s + (6.49 − 6.49i)5-s + (3.26 − 1.15i)6-s + 3.94·7-s + (−1.83 − 7.78i)8-s − 2.99i·9-s + (−17.3 + 6.10i)10-s + (4.31 + 4.31i)11-s + (−6.88 − 0.743i)12-s + (4.06 + 4.06i)13-s + (−7.11 − 3.40i)14-s + 15.9i·15-s + (−3.41 + 15.6i)16-s − 14.5·17-s + ⋯
L(s)  = 1  + (−0.901 − 0.431i)2-s + (−0.408 + 0.408i)3-s + (0.627 + 0.778i)4-s + (1.29 − 1.29i)5-s + (0.544 − 0.191i)6-s + 0.563·7-s + (−0.229 − 0.973i)8-s − 0.333i·9-s + (−1.73 + 0.610i)10-s + (0.391 + 0.391i)11-s + (−0.574 − 0.0619i)12-s + (0.312 + 0.312i)13-s + (−0.508 − 0.243i)14-s + 1.06i·15-s + (−0.213 + 0.976i)16-s − 0.856·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.570i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.820 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $0.820 + 0.570i$
motivic weight  =  \(2\)
character  :  $\chi_{48} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :1),\ 0.820 + 0.570i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.802682 - 0.251676i\)
\(L(\frac12)\)  \(\approx\)  \(0.802682 - 0.251676i\)
\(L(2)\)   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 \( 1 + (1.80 + 0.863i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
good5 \( 1 + (-6.49 + 6.49i)T - 25iT^{2} \)
7 \( 1 - 3.94T + 49T^{2} \)
11 \( 1 + (-4.31 - 4.31i)T + 121iT^{2} \)
13 \( 1 + (-4.06 - 4.06i)T + 169iT^{2} \)
17 \( 1 + 14.5T + 289T^{2} \)
19 \( 1 + (-4.94 + 4.94i)T - 361iT^{2} \)
23 \( 1 + 43.6T + 529T^{2} \)
29 \( 1 + (-25.0 - 25.0i)T + 841iT^{2} \)
31 \( 1 - 32.5iT - 961T^{2} \)
37 \( 1 + (-4.14 + 4.14i)T - 1.36e3iT^{2} \)
41 \( 1 - 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (16.1 + 16.1i)T + 1.84e3iT^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (31.5 - 31.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (49.7 + 49.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-44.4 - 44.4i)T + 3.72e3iT^{2} \)
67 \( 1 + (1.64 - 1.64i)T - 4.48e3iT^{2} \)
71 \( 1 - 24.1T + 5.04e3T^{2} \)
73 \( 1 - 10.7iT - 5.32e3T^{2} \)
79 \( 1 + 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (-42.0 + 42.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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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

−15.89559665767867146524215430812, −14.02190501437789231329548190215, −12.72730822416375301608100705845, −11.71717206419249596510768051600, −10.31247069241421869876252357018, −9.357094070571521104669569021504, −8.437509926924430584523506485421, −6.34446829505170908899301860358, −4.65808041655909513948753146786, −1.64318431323729946503727064285, 2.09422229771337631941299255751, 5.83486076223047143544175054607, 6.57794238394518717753423782922, 8.019604165540918242977519368097, 9.675175312592255847189541040831, 10.64957327401779007875698372688, 11.58056419200418922680884135427, 13.68673348488977545754787644734, 14.37308113958754457893814294537, 15.63330350274993862752635468238

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