Properties

Degree 2
Conductor $ 2^{3} \cdot 3 $
Sign $0.524 - 0.851i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.96i)2-s + 1.73·3-s + (−3.73 − 1.43i)4-s + 2.87i·5-s + (−0.633 + 3.40i)6-s − 10.7i·7-s + (4.19 − 6.81i)8-s + 2.99·9-s + (−5.66 − 1.05i)10-s − 8·11-s + (−6.46 − 2.49i)12-s + 15.7i·13-s + (21.1 + 3.93i)14-s + 4.98i·15-s + (11.8 + 10.7i)16-s − 15.8·17-s + ⋯
L(s)  = 1  + (−0.183 + 0.983i)2-s + 0.577·3-s + (−0.933 − 0.359i)4-s + 0.575i·5-s + (−0.105 + 0.567i)6-s − 1.53i·7-s + (0.524 − 0.851i)8-s + 0.333·9-s + (−0.566 − 0.105i)10-s − 0.727·11-s + (−0.538 − 0.207i)12-s + 1.20i·13-s + (1.50 + 0.280i)14-s + 0.332i·15-s + (0.741 + 0.671i)16-s − 0.932·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $0.524 - 0.851i$
motivic weight  =  \(2\)
character  :  $\chi_{24} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 24,\ (\ :1),\ 0.524 - 0.851i)$
$L(\frac{3}{2})$  $\approx$  $0.784597 + 0.438174i$
$L(\frac12)$  $\approx$  $0.784597 + 0.438174i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (0.366 - 1.96i)T \)
3 \( 1 - 1.73T \)
good5 \( 1 - 2.87iT - 25T^{2} \)
7 \( 1 + 10.7iT - 49T^{2} \)
11 \( 1 + 8T + 121T^{2} \)
13 \( 1 - 15.7iT - 169T^{2} \)
17 \( 1 + 15.8T + 289T^{2} \)
19 \( 1 - 1.07T + 361T^{2} \)
23 \( 1 - 21.4iT - 529T^{2} \)
29 \( 1 + 40.0iT - 841T^{2} \)
31 \( 1 + 9.20iT - 961T^{2} \)
37 \( 1 - 9.97iT - 1.36e3T^{2} \)
41 \( 1 - 51.5T + 1.68e3T^{2} \)
43 \( 1 + 12.7T + 1.84e3T^{2} \)
47 \( 1 + 1.54iT - 2.20e3T^{2} \)
53 \( 1 - 28.5iT - 2.80e3T^{2} \)
59 \( 1 + 11.2T + 3.48e3T^{2} \)
61 \( 1 - 1.54iT - 3.72e3T^{2} \)
67 \( 1 + 43.2T + 4.48e3T^{2} \)
71 \( 1 + 84.4iT - 5.04e3T^{2} \)
73 \( 1 - 105.T + 5.32e3T^{2} \)
79 \( 1 - 73.6iT - 6.24e3T^{2} \)
83 \( 1 - 12.2T + 6.88e3T^{2} \)
89 \( 1 - 33.1T + 7.92e3T^{2} \)
97 \( 1 + 69.1T + 9.40e3T^{2} \)
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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

−17.57154116753396557950094918534, −16.45267067245767059072561184582, −15.23961486911368683205397555358, −13.99366662794992606072480698545, −13.38007144419951117460720073910, −10.80626742570984102236577363067, −9.496348007137509508991465712168, −7.77757615597159208351556357411, −6.74789168780566424944579060693, −4.24261628617026933892896956709, 2.68155907806404530091965479096, 5.14059384744607714226052879422, 8.293193361109677564047267773112, 9.110724250132546308985061125581, 10.70464897985039189470434307052, 12.40726944006768489398401158527, 13.01856626614660602989844745783, 14.74469057313655704085639288379, 16.00850540241678952529086997517, 17.86894800987277571094855865699

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