↑

Properties

Label 2-12-12.11-c13-0-6
Degree $2$
Conductor $12$
Sign $-0.444 - 0.895i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−83.2 − 35.5i)2-s + (1.20e3 + 377. i)3-s + (5.66e3 + 5.91e3i)4-s + 5.35e4i·5-s + (−8.68e4 − 7.42e4i)6-s − 1.33e5i·7-s + (−2.61e5 − 6.93e5i)8-s + (1.30e6 + 9.10e5i)9-s + (1.90e6 − 4.45e6i)10-s − 9.24e6·11-s + (4.59e6 + 9.26e6i)12-s − 2.07e6·13-s + (−4.74e6 + 1.11e7i)14-s + (−2.02e7 + 6.45e7i)15-s + (−2.84e6 + 6.70e7i)16-s + 1.68e8i·17-s + ⋯
L(s)  = 1  + (−0.919 − 0.392i)2-s + (0.954 + 0.299i)3-s + (0.691 + 0.721i)4-s + 1.53i·5-s + (−0.760 − 0.649i)6-s − 0.429i·7-s + (−0.353 − 0.935i)8-s + (0.820 + 0.571i)9-s + (0.601 − 1.41i)10-s − 1.57·11-s + (0.444 + 0.895i)12-s − 0.118·13-s + (−0.168 + 0.394i)14-s + (−0.458 + 1.46i)15-s + (−0.0423 + 0.999i)16-s + 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.444 - 0.895i$
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: \((2,\ 12,\ (\ :13/2),\ -0.444 - 0.895i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.604784 + 0.974964i\)
\(L(\frac12)\) \(\approx\) \(0.604784 + 0.974964i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (83.2 + 35.5i)T \)
3 \( 1 + (-1.20e3 - 377. i)T \)
good5 \( 1 - 5.35e4iT - 1.22e9T^{2} \)
7 \( 1 + 1.33e5iT - 9.68e10T^{2} \)
11 \( 1 + 9.24e6T + 3.45e13T^{2} \)
13 \( 1 + 2.07e6T + 3.02e14T^{2} \)
17 \( 1 - 1.68e8iT - 9.90e15T^{2} \)
19 \( 1 + 1.57e8iT - 4.20e16T^{2} \)
23 \( 1 + 2.49e8T + 5.04e17T^{2} \)
29 \( 1 - 1.99e9iT - 1.02e19T^{2} \)
31 \( 1 - 2.33e9iT - 2.44e19T^{2} \)
37 \( 1 + 1.50e10T + 2.43e20T^{2} \)
41 \( 1 - 1.03e10iT - 9.25e20T^{2} \)
43 \( 1 - 2.02e10iT - 1.71e21T^{2} \)
47 \( 1 + 2.93e10T + 5.46e21T^{2} \)
53 \( 1 + 1.23e11iT - 2.60e22T^{2} \)
59 \( 1 - 3.35e11T + 1.04e23T^{2} \)
61 \( 1 - 5.47e11T + 1.61e23T^{2} \)
67 \( 1 - 4.12e11iT - 5.48e23T^{2} \)
71 \( 1 + 8.08e11T + 1.16e24T^{2} \)
73 \( 1 - 1.20e12T + 1.67e24T^{2} \)
79 \( 1 - 8.42e11iT - 4.66e24T^{2} \)
83 \( 1 - 5.04e12T + 8.87e24T^{2} \)
89 \( 1 + 8.51e12iT - 2.19e25T^{2} \)
97 \( 1 + 2.26e12T + 6.73e25T^{2} \)
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

−17.77260400903349311105407158103, −15.84982208581610235808667255793, −14.78254705576147408168153681852, −13.12161156750137333633638863083, −10.76203068900275367217217101400, −10.18507114528784980525960053249, −8.230758571181144549423520055996, −7.03811110255073070319014967181, −3.42745509763082159692177849137, −2.24954139888364556197047975607, 0.55820027630703465969159687274, 2.26117754886440061820704464686, 5.25378143219485572390424286792, 7.65281500080268369724357384362, 8.665802170546339127019175453021, 9.812855634271698731548350221772, 12.20671176960230243201873090730, 13.65810534819920525294520140458, 15.48344623763851316584955254681, 16.31361188437536563102250617965

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