Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.0359 - 0.999i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (723. + 17.4i)2-s + (−2.86e3 − 3.39e4i)3-s + (5.23e5 + 2.52e4i)4-s + 5.49e6i·5-s + (−1.48e6 − 2.46e7i)6-s + 1.05e8i·7-s + (3.78e8 + 2.74e7i)8-s + (−1.14e9 + 1.94e8i)9-s + (−9.57e7 + 3.97e9i)10-s − 7.09e9·11-s + (−6.43e8 − 1.78e10i)12-s − 6.63e10·13-s + (−1.83e9 + 7.61e10i)14-s + (1.86e11 − 1.57e10i)15-s + (2.73e11 + 2.64e10i)16-s + 6.50e11i·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0240i)2-s + (−0.0840 − 0.996i)3-s + (0.998 + 0.0481i)4-s + 1.25i·5-s + (−0.0600 − 0.998i)6-s + 0.985i·7-s + (0.997 + 0.0721i)8-s + (−0.985 + 0.167i)9-s + (−0.0302 + 1.25i)10-s − 0.907·11-s + (−0.0359 − 0.999i)12-s − 1.73·13-s + (−0.0237 + 0.985i)14-s + (1.25 − 0.105i)15-s + (0.995 + 0.0961i)16-s + 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.0359 - 0.999i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ -0.0359 - 0.999i)$
$L(10)$  $\approx$  $2.511793317$
$L(\frac12)$  $\approx$  $2.511793317$
$L(\frac{21}{2})$   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 \( 1 + (-723. - 17.4i)T \)
3 \( 1 + (2.86e3 + 3.39e4i)T \)
good5 \( 1 - 5.49e6iT - 1.90e13T^{2} \)
7 \( 1 - 1.05e8iT - 1.13e16T^{2} \)
11 \( 1 + 7.09e9T + 6.11e19T^{2} \)
13 \( 1 + 6.63e10T + 1.46e21T^{2} \)
17 \( 1 - 6.50e11iT - 2.39e23T^{2} \)
19 \( 1 - 8.23e11iT - 1.97e24T^{2} \)
23 \( 1 - 1.08e13T + 7.46e25T^{2} \)
29 \( 1 + 9.35e13iT - 6.10e27T^{2} \)
31 \( 1 - 1.40e14iT - 2.16e28T^{2} \)
37 \( 1 - 2.24e14T + 6.24e29T^{2} \)
41 \( 1 - 3.53e14iT - 4.39e30T^{2} \)
43 \( 1 + 1.91e15iT - 1.08e31T^{2} \)
47 \( 1 + 1.54e15T + 5.88e31T^{2} \)
53 \( 1 + 2.30e16iT - 5.77e32T^{2} \)
59 \( 1 + 7.28e16T + 4.42e33T^{2} \)
61 \( 1 + 3.87e16T + 8.34e33T^{2} \)
67 \( 1 - 3.26e17iT - 4.95e34T^{2} \)
71 \( 1 - 1.37e17T + 1.49e35T^{2} \)
73 \( 1 - 2.84e17T + 2.53e35T^{2} \)
79 \( 1 - 3.04e17iT - 1.13e36T^{2} \)
83 \( 1 - 1.54e18T + 2.90e36T^{2} \)
89 \( 1 + 1.94e18iT - 1.09e37T^{2} \)
97 \( 1 - 3.32e17T + 5.60e37T^{2} \)
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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

−15.18127939745174999591918879147, −14.49069574167572928254882692523, −12.93013170541104013177315196612, −11.94757367999841011542038499797, −10.55961924000322895225127515283, −7.80144387217403824138449101198, −6.62940910757103460984632056394, −5.38308800567532671085685450568, −2.91876457177305249631246779087, −2.14629629828158637412255930112, 0.53829853040758446945923904512, 2.84640465004266078888375689197, 4.67369516887414929662770744336, 5.05923229654120571178921634779, 7.42666816342303112925832226979, 9.502454265134908711528094766240, 10.92580688451190492750771804594, 12.42732532320741410637931783499, 13.70347583491791151750332740495, 15.11509165903866829357980687873

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