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Properties

Label 2-12-12.11-c13-0-21
Degree $2$
Conductor $12$
Sign $-0.0998 + 0.995i$
Analytic cond. $12.8677$
Root an. cond. $3.58715$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (89.8 − 11.2i)2-s + (433. − 1.18e3i)3-s + (7.93e3 − 2.02e3i)4-s − 2.49e4i·5-s + (2.55e4 − 1.11e5i)6-s + 7.13e4i·7-s + (6.89e5 − 2.71e5i)8-s + (−1.21e6 − 1.02e6i)9-s + (−2.82e5 − 2.24e6i)10-s − 4.95e6·11-s + (1.03e6 − 1.02e7i)12-s + 1.35e7·13-s + (8.05e5 + 6.40e6i)14-s + (−2.96e7 − 1.08e7i)15-s + (5.88e7 − 3.22e7i)16-s + 5.28e7i·17-s + ⋯
L(s)  = 1  + (0.992 − 0.124i)2-s + (0.343 − 0.939i)3-s + (0.968 − 0.247i)4-s − 0.715i·5-s + (0.223 − 0.974i)6-s + 0.229i·7-s + (0.930 − 0.366i)8-s + (−0.764 − 0.644i)9-s + (−0.0892 − 0.709i)10-s − 0.842·11-s + (0.0998 − 0.995i)12-s + 0.777·13-s + (0.0285 + 0.227i)14-s + (−0.671 − 0.245i)15-s + (0.877 − 0.479i)16-s + 0.531i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0998 + 0.995i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.0998 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.0998 + 0.995i$
Analytic conductor: \(12.8677\)
Root analytic conductor: \(3.58715\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :13/2),\ -0.0998 + 0.995i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.39812 - 2.65082i\)
\(L(\frac12)\) \(\approx\) \(2.39812 - 2.65082i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-89.8 + 11.2i)T \)
3 \( 1 + (-433. + 1.18e3i)T \)
good5 \( 1 + 2.49e4iT - 1.22e9T^{2} \)
7 \( 1 - 7.13e4iT - 9.68e10T^{2} \)
11 \( 1 + 4.95e6T + 3.45e13T^{2} \)
13 \( 1 - 1.35e7T + 3.02e14T^{2} \)
17 \( 1 - 5.28e7iT - 9.90e15T^{2} \)
19 \( 1 + 3.46e8iT - 4.20e16T^{2} \)
23 \( 1 - 4.73e8T + 5.04e17T^{2} \)
29 \( 1 - 6.10e9iT - 1.02e19T^{2} \)
31 \( 1 - 6.26e9iT - 2.44e19T^{2} \)
37 \( 1 - 1.95e10T + 2.43e20T^{2} \)
41 \( 1 - 3.71e10iT - 9.25e20T^{2} \)
43 \( 1 + 2.19e10iT - 1.71e21T^{2} \)
47 \( 1 + 9.42e10T + 5.46e21T^{2} \)
53 \( 1 + 1.89e11iT - 2.60e22T^{2} \)
59 \( 1 - 2.89e11T + 1.04e23T^{2} \)
61 \( 1 - 9.25e10T + 1.61e23T^{2} \)
67 \( 1 - 5.54e10iT - 5.48e23T^{2} \)
71 \( 1 + 8.88e11T + 1.16e24T^{2} \)
73 \( 1 + 3.79e11T + 1.67e24T^{2} \)
79 \( 1 + 8.79e11iT - 4.66e24T^{2} \)
83 \( 1 + 3.05e12T + 8.87e24T^{2} \)
89 \( 1 - 7.30e12iT - 2.19e25T^{2} \)
97 \( 1 + 1.15e13T + 6.73e25T^{2} \)
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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

−16.26979872796876444467156133620, −14.85339020733136016793327827993, −13.30271981733251775200482014737, −12.70427109073282920900833281906, −11.12629727559940164459001832696, −8.603076813574593995932149040991, −6.84196002049980929256168301435, −5.14722665875552153698608868434, −2.92406812273945013671889581083, −1.20037123736601070636437230779, 2.68313066162447576972600744707, 4.06849351904498114358827320857, 5.82576008444030079040449519388, 7.84360400598541483588052517234, 10.20601892420694638113608738931, 11.34532806177858230762963521741, 13.37401598133007160384119213696, 14.54651039865672949673310134263, 15.57427224959949575585371451451, 16.71674698031120669010902061159

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