Properties

Degree 2
Conductor $ 2^{3} \cdot 3 $
Sign $0.774 - 0.632i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 1.46i)2-s − 1.73·3-s + (−0.267 + 3.99i)4-s − 7.98i·5-s + (−2.36 − 2.53i)6-s + 2.13i·7-s + (−6.19 + 5.06i)8-s + 2.99·9-s + (11.6 − 10.9i)10-s − 8·11-s + (0.464 − 6.91i)12-s + 11.6i·13-s + (−3.12 + 2.92i)14-s + 13.8i·15-s + (−15.8 − 2.13i)16-s + 11.8·17-s + ⋯
L(s)  = 1  + (0.683 + 0.730i)2-s − 0.577·3-s + (−0.0669 + 0.997i)4-s − 1.59i·5-s + (−0.394 − 0.421i)6-s + 0.305i·7-s + (−0.774 + 0.632i)8-s + 0.333·9-s + (1.16 − 1.09i)10-s − 0.727·11-s + (0.0386 − 0.576i)12-s + 0.898i·13-s + (−0.223 + 0.208i)14-s + 0.921i·15-s + (−0.991 − 0.133i)16-s + 0.697·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $0.774 - 0.632i$
motivic weight  =  \(2\)
character  :  $\chi_{24} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 24,\ (\ :1),\ 0.774 - 0.632i)$
$L(\frac{3}{2})$  $\approx$  $0.985254 + 0.351206i$
$L(\frac12)$  $\approx$  $0.985254 + 0.351206i$
$L(2)$   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 \( 1 + (-1.36 - 1.46i)T \)
3 \( 1 + 1.73T \)
good5 \( 1 + 7.98iT - 25T^{2} \)
7 \( 1 - 2.13iT - 49T^{2} \)
11 \( 1 + 8T + 121T^{2} \)
13 \( 1 - 11.6iT - 169T^{2} \)
17 \( 1 - 11.8T + 289T^{2} \)
19 \( 1 - 14.9T + 361T^{2} \)
23 \( 1 + 4.27iT - 529T^{2} \)
29 \( 1 - 0.573iT - 841T^{2} \)
31 \( 1 + 57.4iT - 961T^{2} \)
37 \( 1 - 27.6iT - 1.36e3T^{2} \)
41 \( 1 + 31.5T + 1.68e3T^{2} \)
43 \( 1 - 28.7T + 1.84e3T^{2} \)
47 \( 1 - 59.5iT - 2.20e3T^{2} \)
53 \( 1 - 31.3iT - 2.80e3T^{2} \)
59 \( 1 + 52.7T + 3.48e3T^{2} \)
61 \( 1 + 59.5iT - 3.72e3T^{2} \)
67 \( 1 + 84.7T + 4.48e3T^{2} \)
71 \( 1 + 42.4iT - 5.04e3T^{2} \)
73 \( 1 + 5.42T + 5.32e3T^{2} \)
79 \( 1 - 44.6iT - 6.24e3T^{2} \)
83 \( 1 - 67.7T + 6.88e3T^{2} \)
89 \( 1 + 133.T + 7.92e3T^{2} \)
97 \( 1 - 97.1T + 9.40e3T^{2} \)
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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

−17.09365488182585908799684007631, −16.44775431237460235918000820954, −15.48206265842981341922040834870, −13.71617300939241384390984271921, −12.63206147962328451585534254910, −11.74012896545958677224446927740, −9.284699578591150270908648951262, −7.82621438104503750076123410916, −5.78138868873153761470703667285, −4.61158122069529405690835623875, 3.19126847826640312997642897644, 5.55622303442588941650820156498, 7.16093454727449572360219821350, 10.19749434703203209978610563842, 10.76293102666116434721923976255, 12.09426334571972855301389625462, 13.57886089834994631130068568919, 14.70441004541540263996554334350, 15.80553222971354371385550171985, 17.87925667205782554548000445744

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