Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $0.983 + 0.183i$
Motivic weight 18
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 362. i·2-s + (−3.61e3 + 1.93e4i)3-s − 1.31e5·4-s − 3.69e6i·5-s + (−7.00e6 − 1.30e6i)6-s + 3.18e7·7-s − 4.74e7i·8-s + (−3.61e8 − 1.39e8i)9-s + 1.33e9·10-s + 7.02e8i·11-s + (4.73e8 − 2.53e9i)12-s + 8.10e9·13-s + 1.15e10i·14-s + (7.14e10 + 1.33e10i)15-s + 1.71e10·16-s − 1.82e11i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.183 + 0.983i)3-s − 0.500·4-s − 1.89i·5-s + (−0.695 − 0.129i)6-s + 0.788·7-s − 0.353i·8-s + (−0.932 − 0.360i)9-s + 1.33·10-s + 0.297i·11-s + (0.0917 − 0.491i)12-s + 0.764·13-s + 0.557i·14-s + (1.85 + 0.347i)15-s + 0.250·16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(19-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $0.983 + 0.183i$
motivic weight  =  \(18\)
character  :  $\chi_{6} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :9),\ 0.983 + 0.183i)$
$L(\frac{19}{2})$  $\approx$  $1.49968 - 0.138781i$
$L(\frac12)$  $\approx$  $1.49968 - 0.138781i$
$L(10)$   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 \( 1 - 362. iT \)
3 \( 1 + (3.61e3 - 1.93e4i)T \)
good5 \( 1 + 3.69e6iT - 3.81e12T^{2} \)
7 \( 1 - 3.18e7T + 1.62e15T^{2} \)
11 \( 1 - 7.02e8iT - 5.55e18T^{2} \)
13 \( 1 - 8.10e9T + 1.12e20T^{2} \)
17 \( 1 + 1.82e11iT - 1.40e22T^{2} \)
19 \( 1 - 2.50e11T + 1.04e23T^{2} \)
23 \( 1 + 9.32e11iT - 3.24e24T^{2} \)
29 \( 1 + 8.17e12iT - 2.10e26T^{2} \)
31 \( 1 - 2.87e13T + 6.99e26T^{2} \)
37 \( 1 - 3.99e13T + 1.68e28T^{2} \)
41 \( 1 + 3.68e14iT - 1.07e29T^{2} \)
43 \( 1 + 2.11e14T + 2.52e29T^{2} \)
47 \( 1 + 2.53e14iT - 1.25e30T^{2} \)
53 \( 1 + 2.48e14iT - 1.08e31T^{2} \)
59 \( 1 - 1.21e14iT - 7.50e31T^{2} \)
61 \( 1 - 7.62e14T + 1.36e32T^{2} \)
67 \( 1 + 4.78e16T + 7.40e32T^{2} \)
71 \( 1 - 6.41e16iT - 2.10e33T^{2} \)
73 \( 1 - 7.57e16T + 3.46e33T^{2} \)
79 \( 1 - 1.35e16T + 1.43e34T^{2} \)
83 \( 1 - 2.11e17iT - 3.49e34T^{2} \)
89 \( 1 - 1.30e17iT - 1.22e35T^{2} \)
97 \( 1 - 9.26e17T + 5.77e35T^{2} \)
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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

−17.71697900027388773652405488996, −16.52354247257674077270947180205, −15.65534879781288595506477099742, −13.77304833072145531357716590767, −11.83813762989031446986257986990, −9.465325063171447525821599117693, −8.308936127550290206990373150178, −5.35665333584062995755861993809, −4.44192665180808016114122343240, −0.73820954359297864085451684902, 1.60603000496686569773600133234, 3.17749461110040617873476189407, 6.21812247830934507349338685393, 7.929462688212163478526248611833, 10.66843542186751548939622679113, 11.59177586466981742220240934126, 13.60226819982695682237118788294, 14.75024473366228967394132016278, 17.67826506652193396398643415952, 18.48111562740521328819986725744

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