Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
Sign $-0.501 - 0.865i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.455 + 1.94i)2-s + (−1.22 + 1.22i)3-s + (−3.58 + 1.77i)4-s + (−3.40 + 3.40i)5-s + (−2.94 − 1.82i)6-s + 12.1·7-s + (−5.08 − 6.17i)8-s − 2.99i·9-s + (−8.18 − 5.08i)10-s + (9.81 + 9.81i)11-s + (2.22 − 6.56i)12-s + (−7.76 − 7.76i)13-s + (5.51 + 23.6i)14-s − 8.34i·15-s + (9.71 − 12.7i)16-s + 9.73·17-s + ⋯
L(s)  = 1  + (0.227 + 0.973i)2-s + (−0.408 + 0.408i)3-s + (−0.896 + 0.443i)4-s + (−0.681 + 0.681i)5-s + (−0.490 − 0.304i)6-s + 1.73·7-s + (−0.635 − 0.772i)8-s − 0.333i·9-s + (−0.818 − 0.508i)10-s + (0.891 + 0.891i)11-s + (0.185 − 0.546i)12-s + (−0.597 − 0.597i)13-s + (0.394 + 1.68i)14-s − 0.556i·15-s + (0.607 − 0.794i)16-s + 0.572·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $-0.501 - 0.865i$
motivic weight  =  \(2\)
character  :  $\chi_{48} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :1),\ -0.501 - 0.865i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.522536 + 0.906932i\)
\(L(\frac12)\)  \(\approx\)  \(0.522536 + 0.906932i\)
\(L(2)\)   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 \( 1 + (-0.455 - 1.94i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
good5 \( 1 + (3.40 - 3.40i)T - 25iT^{2} \)
7 \( 1 - 12.1T + 49T^{2} \)
11 \( 1 + (-9.81 - 9.81i)T + 121iT^{2} \)
13 \( 1 + (7.76 + 7.76i)T + 169iT^{2} \)
17 \( 1 - 9.73T + 289T^{2} \)
19 \( 1 + (-11.2 + 11.2i)T - 361iT^{2} \)
23 \( 1 + 20.2T + 529T^{2} \)
29 \( 1 + (16.4 + 16.4i)T + 841iT^{2} \)
31 \( 1 + 26.3iT - 961T^{2} \)
37 \( 1 + (23.7 - 23.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 24.7iT - 1.68e3T^{2} \)
43 \( 1 + (-29.8 - 29.8i)T + 1.84e3iT^{2} \)
47 \( 1 + 31.3iT - 2.20e3T^{2} \)
53 \( 1 + (-36.8 + 36.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (14.1 + 14.1i)T + 3.48e3iT^{2} \)
61 \( 1 + (42.5 + 42.5i)T + 3.72e3iT^{2} \)
67 \( 1 + (-48.7 + 48.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 7.73T + 5.04e3T^{2} \)
73 \( 1 - 85.4iT - 5.32e3T^{2} \)
79 \( 1 - 105. iT - 6.24e3T^{2} \)
83 \( 1 + (62.1 - 62.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 127. iT - 7.92e3T^{2} \)
97 \( 1 + 147.T + 9.40e3T^{2} \)
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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

−15.43867488704754338035928274037, −14.91304078762027893065935441593, −14.09655152448583194537013843331, −12.18805912443162290070164843715, −11.32286415329425566198341477206, −9.729414536731168307688916640181, −8.048248649426349862430703702616, −7.15309427592625951540581974727, −5.32194229187139031195679806999, −4.10867673311570854249237111966, 1.38414053555612297403574408004, 4.18875414088002136895591747798, 5.44705190007040149561065731620, 7.85518945540525527385235323285, 8.926500160787073118317282029120, 10.76003845647703277546485909721, 11.87619527306698280876188132468, 12.11794158803237915818573461782, 13.94205703712406635852074249465, 14.53629879934108302577156810693

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