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Properties

Label 2-12-12.11-c13-0-4
Degree $2$
Conductor $12$
Sign $-0.128 - 0.991i$
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  + (−63.1 + 64.8i)2-s + (−1.25e3 + 128. i)3-s + (−218. − 8.18e3i)4-s − 1.78e3i·5-s + (7.09e4 − 8.95e4i)6-s − 3.16e5i·7-s + (5.44e5 + 5.02e5i)8-s + (1.56e6 − 3.22e5i)9-s + (1.16e5 + 1.12e5i)10-s − 5.24e6·11-s + (1.32e6 + 1.02e7i)12-s − 7.16e6·13-s + (2.05e7 + 2.00e7i)14-s + (2.29e5 + 2.24e6i)15-s + (−6.70e7 + 3.57e6i)16-s + 1.30e8i·17-s + ⋯
L(s)  = 1  + (−0.697 + 0.716i)2-s + (−0.994 + 0.101i)3-s + (−0.0266 − 0.999i)4-s − 0.0511i·5-s + (0.621 − 0.783i)6-s − 1.01i·7-s + (0.734 + 0.678i)8-s + (0.979 − 0.202i)9-s + (0.0366 + 0.0357i)10-s − 0.892·11-s + (0.128 + 0.991i)12-s − 0.411·13-s + (0.729 + 0.710i)14-s + (0.00521 + 0.0509i)15-s + (−0.998 + 0.0533i)16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.128 - 0.991i$
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: \((2,\ 12,\ (\ :13/2),\ -0.128 - 0.991i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.400678 + 0.455858i\)
\(L(\frac12)\) \(\approx\) \(0.400678 + 0.455858i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (63.1 - 64.8i)T \)
3 \( 1 + (1.25e3 - 128. i)T \)
good5 \( 1 + 1.78e3iT - 1.22e9T^{2} \)
7 \( 1 + 3.16e5iT - 9.68e10T^{2} \)
11 \( 1 + 5.24e6T + 3.45e13T^{2} \)
13 \( 1 + 7.16e6T + 3.02e14T^{2} \)
17 \( 1 - 1.30e8iT - 9.90e15T^{2} \)
19 \( 1 - 1.09e8iT - 4.20e16T^{2} \)
23 \( 1 - 5.41e8T + 5.04e17T^{2} \)
29 \( 1 + 4.73e9iT - 1.02e19T^{2} \)
31 \( 1 - 8.25e9iT - 2.44e19T^{2} \)
37 \( 1 - 1.92e10T + 2.43e20T^{2} \)
41 \( 1 - 4.37e10iT - 9.25e20T^{2} \)
43 \( 1 - 6.99e10iT - 1.71e21T^{2} \)
47 \( 1 + 3.55e10T + 5.46e21T^{2} \)
53 \( 1 + 7.53e10iT - 2.60e22T^{2} \)
59 \( 1 - 2.47e11T + 1.04e23T^{2} \)
61 \( 1 + 5.16e11T + 1.61e23T^{2} \)
67 \( 1 + 1.50e11iT - 5.48e23T^{2} \)
71 \( 1 - 3.92e11T + 1.16e24T^{2} \)
73 \( 1 - 5.99e11T + 1.67e24T^{2} \)
79 \( 1 + 1.31e11iT - 4.66e24T^{2} \)
83 \( 1 - 1.71e12T + 8.87e24T^{2} \)
89 \( 1 - 3.37e12iT - 2.19e25T^{2} \)
97 \( 1 - 1.18e13T + 6.73e25T^{2} \)
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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

−17.13090736112566700183540447067, −16.32083278302302067406166559545, −14.88020107100566030388780219438, −13.03254292710736696557833089547, −10.93484903598392406138061944658, −10.01011760393871941122878759019, −7.82549261836104068664576829765, −6.38456011943808123017518291490, −4.75239741855318348813945891846, −1.05306758741866733891664645521, 0.46766914607073343576304526274, 2.51178149499310618657739991373, 5.09914948119258038255240090401, 7.25210907879635989919084918455, 9.197958455279945911036013561683, 10.74467240036406774481685505348, 11.91091480926528384672219262412, 13.02250001921348022440168984152, 15.60390544493932538015856361577, 16.82495015096545666850645447967

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