↑

Properties

Label 2-12-12.11-c9-0-9
Degree $2$
Conductor $12$
Sign $0.895 + 0.444i$
Analytic cond. $6.18043$
Root an. cond. $2.48604$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−8.73 + 20.8i)2-s + (132. − 45.7i)3-s + (−359. − 364. i)4-s − 1.83e3i·5-s + (−202. + 3.16e3i)6-s − 1.30e3i·7-s + (1.07e4 − 4.31e3i)8-s + (1.54e4 − 1.21e4i)9-s + (3.82e4 + 1.60e4i)10-s + 6.70e4·11-s + (−6.43e4 − 3.19e4i)12-s − 1.28e5·13-s + (2.72e4 + 1.13e4i)14-s + (−8.38e4 − 2.42e5i)15-s + (−3.89e3 + 2.62e5i)16-s + 7.87e4i·17-s + ⋯
L(s)  = 1  + (−0.386 + 0.922i)2-s + (0.945 − 0.326i)3-s + (−0.701 − 0.712i)4-s − 1.31i·5-s + (−0.0639 + 0.997i)6-s − 0.205i·7-s + (0.928 − 0.372i)8-s + (0.787 − 0.616i)9-s + (1.20 + 0.506i)10-s + 1.38·11-s + (−0.895 − 0.444i)12-s − 1.24·13-s + (0.189 + 0.0792i)14-s + (−0.427 − 1.23i)15-s + (−0.0148 + 0.999i)16-s + 0.228i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(6.18043\)
Root analytic conductor: \(2.48604\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :9/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.60360 - 0.375816i\)
\(L(\frac12)\) \(\approx\) \(1.60360 - 0.375816i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8.73 - 20.8i)T \)
3 \( 1 + (-132. + 45.7i)T \)
good5 \( 1 + 1.83e3iT - 1.95e6T^{2} \)
7 \( 1 + 1.30e3iT - 4.03e7T^{2} \)
11 \( 1 - 6.70e4T + 2.35e9T^{2} \)
13 \( 1 + 1.28e5T + 1.06e10T^{2} \)
17 \( 1 - 7.87e4iT - 1.18e11T^{2} \)
19 \( 1 + 7.05e5iT - 3.22e11T^{2} \)
23 \( 1 - 3.02e5T + 1.80e12T^{2} \)
29 \( 1 + 1.71e5iT - 1.45e13T^{2} \)
31 \( 1 - 8.57e6iT - 2.64e13T^{2} \)
37 \( 1 + 2.65e6T + 1.29e14T^{2} \)
41 \( 1 - 3.10e7iT - 3.27e14T^{2} \)
43 \( 1 - 5.87e6iT - 5.02e14T^{2} \)
47 \( 1 - 3.06e7T + 1.11e15T^{2} \)
53 \( 1 + 6.60e4iT - 3.29e15T^{2} \)
59 \( 1 - 3.52e7T + 8.66e15T^{2} \)
61 \( 1 + 8.17e7T + 1.16e16T^{2} \)
67 \( 1 - 4.25e7iT - 2.72e16T^{2} \)
71 \( 1 + 7.59e7T + 4.58e16T^{2} \)
73 \( 1 - 1.16e8T + 5.88e16T^{2} \)
79 \( 1 - 4.69e8iT - 1.19e17T^{2} \)
83 \( 1 - 4.67e8T + 1.86e17T^{2} \)
89 \( 1 - 3.29e7iT - 3.50e17T^{2} \)
97 \( 1 + 1.39e9T + 7.60e17T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.61200688290995027099598105769, −16.60143611045574559513355567195, −15.09397641666197251385627093447, −13.88664944638309021882694266894, −12.52367568862970998935254589028, −9.504985293645826021855482563665, −8.621219921583940921439209274769, −7.00861600810562028611796509909, −4.60020565203827896482042411218, −1.10913969595635650550674298413, 2.28762276454931490630696937093, 3.79008496721005422897180912810, 7.42665779778737207693828863402, 9.267248701440558106638095356004, 10.42315838222142110991881866349, 12.04676821827502735261233289631, 14.00138748495780356222634315024, 14.86770222887867911049694163399, 17.01221072234116249692532606048, 18.67350406758531605360230431359

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