Properties

Degree 2
Conductor $ 2^{3} \cdot 3 $
Sign $-0.408 - 0.912i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 + 10.1i)2-s + (−9 − 45.8i)3-s + (−79.9 − 99.9i)4-s + 97.9·5-s + (512. + 133. i)6-s + 1.54e3i·7-s + (1.41e3 − 326. i)8-s + (−2.02e3 + 826. i)9-s + (−479. + 999. i)10-s − 1.78e3i·11-s + (−3.86e3 + 4.57e3i)12-s + 1.01e4i·13-s + (−1.57e4 − 7.58e3i)14-s + (−881. − 4.49e3i)15-s + (−3.58e3 + 1.59e4i)16-s + 2.75e4i·17-s + ⋯
L(s)  = 1  + (−0.433 + 0.901i)2-s + (−0.192 − 0.981i)3-s + (−0.624 − 0.780i)4-s + 0.350·5-s + (0.967 + 0.251i)6-s + 1.70i·7-s + (0.974 − 0.225i)8-s + (−0.925 + 0.377i)9-s + (−0.151 + 0.315i)10-s − 0.404i·11-s + (−0.645 + 0.763i)12-s + 1.28i·13-s + (−1.53 − 0.738i)14-s + (−0.0674 − 0.343i)15-s + (−0.218 + 0.975i)16-s + 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $-0.408 - 0.912i$
motivic weight  =  \(7\)
character  :  $\chi_{24} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 24,\ (\ :7/2),\ -0.408 - 0.912i)$
$L(4)$  $\approx$  $0.502367 + 0.775340i$
$L(\frac12)$  $\approx$  $0.502367 + 0.775340i$
$L(\frac{9}{2})$   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 \( 1 + (4.89 - 10.1i)T \)
3 \( 1 + (9 + 45.8i)T \)
good5 \( 1 - 97.9T + 7.81e4T^{2} \)
7 \( 1 - 1.54e3iT - 8.23e5T^{2} \)
11 \( 1 + 1.78e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.01e4iT - 6.27e7T^{2} \)
17 \( 1 - 2.75e4iT - 4.10e8T^{2} \)
19 \( 1 - 1.15e4T + 8.93e8T^{2} \)
23 \( 1 - 5.55e4T + 3.40e9T^{2} \)
29 \( 1 + 5.47e4T + 1.72e10T^{2} \)
31 \( 1 + 7.16e4iT - 2.75e10T^{2} \)
37 \( 1 - 2.82e5iT - 9.49e10T^{2} \)
41 \( 1 + 4.03e5iT - 1.94e11T^{2} \)
43 \( 1 - 4.95e5T + 2.71e11T^{2} \)
47 \( 1 + 1.13e6T + 5.06e11T^{2} \)
53 \( 1 + 5.72e5T + 1.17e12T^{2} \)
59 \( 1 + 1.44e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.78e6iT - 3.14e12T^{2} \)
67 \( 1 - 1.40e6T + 6.06e12T^{2} \)
71 \( 1 - 3.56e6T + 9.09e12T^{2} \)
73 \( 1 + 2.22e6T + 1.10e13T^{2} \)
79 \( 1 - 5.59e6iT - 1.92e13T^{2} \)
83 \( 1 - 3.01e6iT - 2.71e13T^{2} \)
89 \( 1 - 5.84e6iT - 4.42e13T^{2} \)
97 \( 1 - 6.86e6T + 8.07e13T^{2} \)
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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

−16.71354490616480791624222677618, −15.30825677947823901226257622517, −14.14752024749528118981410526518, −12.84598674040532695324644180378, −11.42075132100137735024749145402, −9.276440927987274633659092311121, −8.251205567555538711877387299895, −6.50417438138518906960741438036, −5.54449617137690454772694594239, −1.83175964532965028086829800452, 0.59242553597870120860012054756, 3.31882280413394818242892203812, 4.83073572100013523590809034938, 7.58121902724244265860583281121, 9.494858364926015074978772945243, 10.34253788759327420059655150819, 11.31614524529457352486230619775, 13.09979798313144126731631793538, 14.24948819525987632942398246125, 16.08037274816405547863935839035

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