Properties

Degree 2
Conductor $ 2^{3} \cdot 3 $
Sign $0.00418 + 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.87i)2-s + (−1.41 − 2.64i)3-s + (−3 + 2.64i)4-s + 5.65·5-s + (−3.94 + 4.51i)6-s + 4·7-s + (7.07 + 3.74i)8-s + (−5 + 7.48i)9-s + (−4.00 − 10.5i)10-s − 8.48·11-s + (11.2 + 4.19i)12-s + 10.5i·13-s + (−2.82 − 7.48i)14-s + (−8.00 − 14.9i)15-s + (1.99 − 15.8i)16-s − 14.9i·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s + (−0.471 − 0.881i)3-s + (−0.750 + 0.661i)4-s + 1.13·5-s + (−0.658 + 0.752i)6-s + 0.571·7-s + (0.883 + 0.467i)8-s + (−0.555 + 0.831i)9-s + (−0.400 − 1.05i)10-s − 0.771·11-s + (0.936 + 0.349i)12-s + 0.814i·13-s + (−0.202 − 0.534i)14-s + (−0.533 − 0.997i)15-s + (0.124 − 0.992i)16-s − 0.880i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00418 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00418 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $0.00418 + 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{24} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 24,\ (\ :1),\ 0.00418 + 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.534844 - 0.532608i\)
\(L(\frac12)\)  \(\approx\)  \(0.534844 - 0.532608i\)
\(L(2)\)   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 \( 1 + (0.707 + 1.87i)T \)
3 \( 1 + (1.41 + 2.64i)T \)
good5 \( 1 - 5.65T + 25T^{2} \)
7 \( 1 - 4T + 49T^{2} \)
11 \( 1 + 8.48T + 121T^{2} \)
13 \( 1 - 10.5iT - 169T^{2} \)
17 \( 1 + 14.9iT - 289T^{2} \)
19 \( 1 - 5.29iT - 361T^{2} \)
23 \( 1 - 29.9iT - 529T^{2} \)
29 \( 1 + 16.9T + 841T^{2} \)
31 \( 1 + 4T + 961T^{2} \)
37 \( 1 + 52.9iT - 1.36e3T^{2} \)
41 \( 1 + 29.9iT - 1.68e3T^{2} \)
43 \( 1 + 5.29iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 50.9T + 2.80e3T^{2} \)
59 \( 1 + 48.0T + 3.48e3T^{2} \)
61 \( 1 - 95.2iT - 3.72e3T^{2} \)
67 \( 1 + 47.6iT - 4.48e3T^{2} \)
71 \( 1 + 89.7iT - 5.04e3T^{2} \)
73 \( 1 + 6T + 5.32e3T^{2} \)
79 \( 1 - 124T + 6.24e3T^{2} \)
83 \( 1 - 2.82T + 6.88e3T^{2} \)
89 \( 1 - 104. iT - 7.92e3T^{2} \)
97 \( 1 - 118T + 9.40e3T^{2} \)
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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

−17.73030614622155761934021313832, −16.59151766647779599834726882434, −14.01171933621875929984920208097, −13.30585491727059322223027458989, −11.94041167474267908976916677818, −10.79036843865912096372979466727, −9.292175672440491200167963865656, −7.55293857840674426208737803386, −5.38400542037203543276090817067, −1.99519762672624888677905786603, 4.98220427408142433639761568124, 6.14447145950804349033210209199, 8.351408340978118145097745896039, 9.846670408141429861209173533844, 10.72062752973857449709958500144, 13.06787330122284560108302223006, 14.49757130445901706271734898481, 15.42479459457495621408542591918, 16.75613951283113103331764162205, 17.55015296633442841629898496836

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