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Properties

Label 2-12-4.3-c20-0-9
Degree $2$
Conductor $12$
Sign $0.947 + 0.318i$
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  + (−831. + 597. i)2-s + 3.40e4i·3-s + (3.33e5 − 9.93e5i)4-s − 5.52e6·5-s + (−2.03e7 − 2.83e7i)6-s + 2.40e8i·7-s + (3.16e8 + 1.02e9i)8-s − 1.16e9·9-s + (4.59e9 − 3.30e9i)10-s − 1.17e10i·11-s + (3.38e10 + 1.13e10i)12-s − 1.31e11·13-s + (−1.43e11 − 2.00e11i)14-s − 1.88e11i·15-s + (−8.76e11 − 6.63e11i)16-s − 1.84e12·17-s + ⋯
L(s)  = 1  + (−0.811 + 0.583i)2-s + 0.577i·3-s + (0.318 − 0.947i)4-s − 0.565·5-s + (−0.337 − 0.468i)6-s + 0.852i·7-s + (0.294 + 0.955i)8-s − 0.333·9-s + (0.459 − 0.330i)10-s − 0.452i·11-s + (0.547 + 0.183i)12-s − 0.956·13-s + (−0.497 − 0.692i)14-s − 0.326i·15-s + (−0.797 − 0.603i)16-s − 0.917·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.947 + 0.318i$
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: \((2,\ 12,\ (\ :10),\ 0.947 + 0.318i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.5788798553\)
\(L(\frac12)\) \(\approx\) \(0.5788798553\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (831. - 597. i)T \)
3 \( 1 - 3.40e4iT \)
good5 \( 1 + 5.52e6T + 9.53e13T^{2} \)
7 \( 1 - 2.40e8iT - 7.97e16T^{2} \)
11 \( 1 + 1.17e10iT - 6.72e20T^{2} \)
13 \( 1 + 1.31e11T + 1.90e22T^{2} \)
17 \( 1 + 1.84e12T + 4.06e24T^{2} \)
19 \( 1 - 3.34e12iT - 3.75e25T^{2} \)
23 \( 1 + 5.78e13iT - 1.71e27T^{2} \)
29 \( 1 - 6.89e14T + 1.76e29T^{2} \)
31 \( 1 - 3.48e14iT - 6.71e29T^{2} \)
37 \( 1 - 3.15e15T + 2.31e31T^{2} \)
41 \( 1 - 1.86e16T + 1.80e32T^{2} \)
43 \( 1 + 1.40e15iT - 4.67e32T^{2} \)
47 \( 1 - 1.07e15iT - 2.76e33T^{2} \)
53 \( 1 + 1.68e16T + 3.05e34T^{2} \)
59 \( 1 + 6.06e17iT - 2.61e35T^{2} \)
61 \( 1 + 7.66e17T + 5.08e35T^{2} \)
67 \( 1 - 1.19e18iT - 3.32e36T^{2} \)
71 \( 1 + 4.68e18iT - 1.05e37T^{2} \)
73 \( 1 - 8.31e18T + 1.84e37T^{2} \)
79 \( 1 + 1.79e19iT - 8.96e37T^{2} \)
83 \( 1 + 1.72e19iT - 2.40e38T^{2} \)
89 \( 1 - 1.92e19T + 9.72e38T^{2} \)
97 \( 1 - 1.24e19T + 5.43e39T^{2} \)
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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

−15.46445275476083396358610294388, −14.38304962775871939854173429721, −12.01217822283055804264347289751, −10.60015395244247195265638310121, −9.171308409403969431911719503422, −8.042405512474327164205883512444, −6.26971084332047987632922972676, −4.71278522680563908201805132158, −2.47098620518257125993956655470, −0.30577876974230038854912842038, 0.917817439260486554965523739837, 2.48731579554999205991814692277, 4.20964130318442780968337751971, 6.97898397414258041641057928171, 7.88542607399243592715243685617, 9.579167476492568918776907389393, 11.08001171990035399511548350652, 12.23895211975193968148058711156, 13.57383096939516522449803416327, 15.56832300690954655631939026038

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