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Properties

Label 2-12-4.3-c20-0-19
Degree $2$
Conductor $12$
Sign $-0.999 - 0.0296i$
Analytic cond. $30.4216$
Root an. cond. $5.51558$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (734. − 713. i)2-s − 3.40e4i·3-s + (3.10e4 − 1.04e6i)4-s + 6.80e6·5-s + (−2.43e7 − 2.50e7i)6-s − 4.46e7i·7-s + (−7.24e8 − 7.92e8i)8-s − 1.16e9·9-s + (5.00e9 − 4.85e9i)10-s − 3.23e10i·11-s + (−3.57e10 − 1.05e9i)12-s + 7.05e10·13-s + (−3.18e10 − 3.27e10i)14-s − 2.32e11i·15-s + (−1.09e12 − 6.51e10i)16-s − 1.15e12·17-s + ⋯
L(s)  = 1  + (0.717 − 0.696i)2-s − 0.577i·3-s + (0.0296 − 0.999i)4-s + 0.696·5-s + (−0.402 − 0.414i)6-s − 0.157i·7-s + (−0.674 − 0.737i)8-s − 0.333·9-s + (0.500 − 0.485i)10-s − 1.24i·11-s + (−0.577 − 0.0171i)12-s + 0.511·13-s + (−0.109 − 0.113i)14-s − 0.402i·15-s + (−0.998 − 0.0592i)16-s − 0.573·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0296i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.999 - 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.999 - 0.0296i$
Analytic conductor: \(30.4216\)
Root analytic conductor: \(5.51558\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :10),\ -0.999 - 0.0296i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(2.662595667\)
\(L(\frac12)\) \(\approx\) \(2.662595667\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-734. + 713. i)T \)
3 \( 1 + 3.40e4iT \)
good5 \( 1 - 6.80e6T + 9.53e13T^{2} \)
7 \( 1 + 4.46e7iT - 7.97e16T^{2} \)
11 \( 1 + 3.23e10iT - 6.72e20T^{2} \)
13 \( 1 - 7.05e10T + 1.90e22T^{2} \)
17 \( 1 + 1.15e12T + 4.06e24T^{2} \)
19 \( 1 + 4.69e12iT - 3.75e25T^{2} \)
23 \( 1 - 3.62e13iT - 1.71e27T^{2} \)
29 \( 1 - 3.47e14T + 1.76e29T^{2} \)
31 \( 1 - 3.23e14iT - 6.71e29T^{2} \)
37 \( 1 + 6.74e15T + 2.31e31T^{2} \)
41 \( 1 + 2.06e16T + 1.80e32T^{2} \)
43 \( 1 + 2.47e16iT - 4.67e32T^{2} \)
47 \( 1 - 4.97e16iT - 2.76e33T^{2} \)
53 \( 1 - 2.42e17T + 3.05e34T^{2} \)
59 \( 1 + 2.94e17iT - 2.61e35T^{2} \)
61 \( 1 - 1.13e18T + 5.08e35T^{2} \)
67 \( 1 + 2.38e18iT - 3.32e36T^{2} \)
71 \( 1 + 3.37e18iT - 1.05e37T^{2} \)
73 \( 1 - 3.65e18T + 1.84e37T^{2} \)
79 \( 1 - 1.94e18iT - 8.96e37T^{2} \)
83 \( 1 + 2.58e19iT - 2.40e38T^{2} \)
89 \( 1 - 4.32e19T + 9.72e38T^{2} \)
97 \( 1 - 2.46e19T + 5.43e39T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.84421534617884733828965964424, −13.42871885858696419074128178012, −11.79245098285294860233912831780, −10.56647947217688252975297982795, −8.870220187534256525217959692943, −6.58266114958050016724208531323, −5.38122862891094028156555312486, −3.41284116708826955032640145068, −1.94775204140556704206970705647, −0.62423074769147676802873371625, 2.24532211488944908523026065489, 4.02499476858179114253517329899, 5.34369706874359450928062023287, 6.72430220875135829363161722837, 8.552515250677263680519739531844, 10.11489462890892109311791620231, 12.01884466677098925099862496959, 13.41661010494534262208299325892, 14.66992781598114013112920485578, 15.72903744877654187420158045363

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