Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.964 - 0.262i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (627. − 362. i)2-s + (−8.68e3 − 3.29e4i)3-s + (2.61e5 − 4.54e5i)4-s − 6.03e6i·5-s + (−1.73e7 − 1.75e7i)6-s − 9.28e7i·7-s + (−2.07e5 − 3.79e8i)8-s + (−1.01e9 + 5.72e8i)9-s + (−2.18e9 − 3.78e9i)10-s + 6.13e9·11-s + (−1.72e10 − 4.69e9i)12-s + 4.99e10·13-s + (−3.36e10 − 5.82e10i)14-s + (−1.98e11 + 5.23e10i)15-s + (−1.37e11 − 2.37e11i)16-s + 2.01e11i·17-s + ⋯
L(s)  = 1  + (0.865 − 0.500i)2-s + (−0.254 − 0.967i)3-s + (0.499 − 0.866i)4-s − 1.38i·5-s + (−0.704 − 0.709i)6-s − 0.870i·7-s + (−0.000545 − 0.999i)8-s + (−0.870 + 0.492i)9-s + (−0.690 − 1.19i)10-s + 0.784·11-s + (−0.964 − 0.262i)12-s + 1.30·13-s + (−0.435 − 0.753i)14-s + (−1.33 + 0.351i)15-s + (−0.500 − 0.865i)16-s + 0.412i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.964 - 0.262i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ -0.964 - 0.262i)$
$L(10)$  $\approx$  $3.119837508$
$L(\frac12)$  $\approx$  $3.119837508$
$L(\frac{21}{2})$   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 \( 1 + (-627. + 362. i)T \)
3 \( 1 + (8.68e3 + 3.29e4i)T \)
good5 \( 1 + 6.03e6iT - 1.90e13T^{2} \)
7 \( 1 + 9.28e7iT - 1.13e16T^{2} \)
11 \( 1 - 6.13e9T + 6.11e19T^{2} \)
13 \( 1 - 4.99e10T + 1.46e21T^{2} \)
17 \( 1 - 2.01e11iT - 2.39e23T^{2} \)
19 \( 1 - 4.67e11iT - 1.97e24T^{2} \)
23 \( 1 - 8.46e12T + 7.46e25T^{2} \)
29 \( 1 - 2.39e13iT - 6.10e27T^{2} \)
31 \( 1 - 1.41e14iT - 2.16e28T^{2} \)
37 \( 1 + 1.37e15T + 6.24e29T^{2} \)
41 \( 1 + 3.22e15iT - 4.39e30T^{2} \)
43 \( 1 - 5.84e15iT - 1.08e31T^{2} \)
47 \( 1 - 1.11e16T + 5.88e31T^{2} \)
53 \( 1 + 3.17e16iT - 5.77e32T^{2} \)
59 \( 1 + 1.59e16T + 4.42e33T^{2} \)
61 \( 1 - 1.40e17T + 8.34e33T^{2} \)
67 \( 1 + 3.36e16iT - 4.95e34T^{2} \)
71 \( 1 - 6.45e17T + 1.49e35T^{2} \)
73 \( 1 + 6.40e17T + 2.53e35T^{2} \)
79 \( 1 - 3.85e17iT - 1.13e36T^{2} \)
83 \( 1 + 5.19e17T + 2.90e36T^{2} \)
89 \( 1 - 5.87e17iT - 1.09e37T^{2} \)
97 \( 1 + 4.81e18T + 5.60e37T^{2} \)
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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

−14.09119265612013805919027182090, −13.13822376727379907991734362323, −12.22419037110483335106751764917, −10.86485943180467125628652171135, −8.721865676685334230652256838920, −6.78357541031243186011081926656, −5.33096367457235103545153450898, −3.79335996170883689527375605426, −1.48191825354996702275956116451, −0.866426325587827530949535251757, 2.75027389983218179769141512752, 3.85700398127329986139431448498, 5.63369504835101727513133097003, 6.74071853460815051973894772901, 8.868711619704171765363012900153, 10.85485951270756812579807748851, 11.78532980158410473438823879702, 13.87354663422915734920991736751, 15.02197392392257604253896954089, 15.68770447074218043932616178501

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