Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.831 - 0.555i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (519. − 504. i)2-s + (1.97e4 + 2.77e4i)3-s + (1.55e4 − 5.24e5i)4-s − 1.34e6i·5-s + (2.42e7 + 4.44e6i)6-s + 1.72e8i·7-s + (−2.56e8 − 2.80e8i)8-s + (−3.79e8 + 1.09e9i)9-s + (−6.78e8 − 6.98e8i)10-s + 1.32e10·11-s + (1.48e10 − 9.93e9i)12-s − 2.84e10·13-s + (8.70e10 + 8.96e10i)14-s + (3.73e10 − 2.66e10i)15-s + (−2.74e11 − 1.62e10i)16-s + 1.96e11i·17-s + ⋯
L(s)  = 1  + (0.717 − 0.696i)2-s + (0.580 + 0.814i)3-s + (0.0295 − 0.999i)4-s − 0.307i·5-s + (0.983 + 0.180i)6-s + 1.61i·7-s + (−0.675 − 0.737i)8-s + (−0.326 + 0.945i)9-s + (−0.214 − 0.220i)10-s + 1.70·11-s + (0.831 − 0.555i)12-s − 0.745·13-s + (1.12 + 1.15i)14-s + (0.250 − 0.178i)15-s + (−0.998 − 0.0591i)16-s + 0.401i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.831 - 0.555i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ 0.831 - 0.555i)$
$L(10)$  $\approx$  $3.508596092$
$L(\frac12)$  $\approx$  $3.508596092$
$L(\frac{21}{2})$   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 \( 1 + (-519. + 504. i)T \)
3 \( 1 + (-1.97e4 - 2.77e4i)T \)
good5 \( 1 + 1.34e6iT - 1.90e13T^{2} \)
7 \( 1 - 1.72e8iT - 1.13e16T^{2} \)
11 \( 1 - 1.32e10T + 6.11e19T^{2} \)
13 \( 1 + 2.84e10T + 1.46e21T^{2} \)
17 \( 1 - 1.96e11iT - 2.39e23T^{2} \)
19 \( 1 - 2.26e12iT - 1.97e24T^{2} \)
23 \( 1 - 9.34e12T + 7.46e25T^{2} \)
29 \( 1 + 1.66e13iT - 6.10e27T^{2} \)
31 \( 1 + 4.90e13iT - 2.16e28T^{2} \)
37 \( 1 - 9.66e14T + 6.24e29T^{2} \)
41 \( 1 - 1.16e15iT - 4.39e30T^{2} \)
43 \( 1 - 3.53e14iT - 1.08e31T^{2} \)
47 \( 1 + 2.94e15T + 5.88e31T^{2} \)
53 \( 1 + 9.59e15iT - 5.77e32T^{2} \)
59 \( 1 - 1.43e15T + 4.42e33T^{2} \)
61 \( 1 + 1.36e16T + 8.34e33T^{2} \)
67 \( 1 + 3.13e17iT - 4.95e34T^{2} \)
71 \( 1 + 4.79e17T + 1.49e35T^{2} \)
73 \( 1 + 3.71e17T + 2.53e35T^{2} \)
79 \( 1 + 5.54e17iT - 1.13e36T^{2} \)
83 \( 1 + 1.99e18T + 2.90e36T^{2} \)
89 \( 1 + 2.47e18iT - 1.09e37T^{2} \)
97 \( 1 + 5.09e18T + 5.60e37T^{2} \)
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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

−15.00087838347816478718769770042, −14.52341832075352929996233125278, −12.61282059987894978278562786856, −11.53313456031944535356342275951, −9.725772377839648738969274685527, −8.804815358372884112292356461169, −5.92113565069519960003354959781, −4.54082284188372693689218475534, −3.10226650012275257202205042275, −1.71961640215708260234831617054, 0.887425879770882160475334759139, 2.99388387121091578368074904860, 4.36916096355967958678291039862, 6.80584775157536947262843187830, 7.20431787944548775488078034241, 9.055122556559390637799471788260, 11.44582186346067906064456404416, 12.99298526509404278970614879413, 14.04374708740957631484146070244, 14.80831428179404719447209815912

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