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Properties

Label 2-12-12.11-c7-0-8
Degree $2$
Conductor $12$
Sign $0.999 - 0.0283i$
Analytic cond. $3.74862$
Root an. cond. $1.93613$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.3 + 4.47i)2-s + (31.1 − 34.8i)3-s + (88 + 92.9i)4-s − 205. i·5-s + (479. − 222. i)6-s + 999. i·7-s + (498. + 1.35e3i)8-s + (−242. − 2.17e3i)9-s + (920 − 2.13e3i)10-s − 1.97e3·11-s + (5.98e3 − 169. i)12-s − 1.27e4·13-s + (−4.46e3 + 1.03e4i)14-s + (−7.17e3 − 6.41e3i)15-s + (−896. + 1.63e4i)16-s − 1.88e4i·17-s + ⋯
L(s)  = 1  + (0.918 + 0.395i)2-s + (0.666 − 0.745i)3-s + (0.687 + 0.726i)4-s − 0.735i·5-s + (0.907 − 0.421i)6-s + 1.10i·7-s + (0.344 + 0.938i)8-s + (−0.111 − 0.993i)9-s + (0.290 − 0.676i)10-s − 0.447·11-s + (0.999 − 0.0283i)12-s − 1.60·13-s + (−0.435 + 1.01i)14-s + (−0.548 − 0.490i)15-s + (−0.0546 + 0.998i)16-s − 0.930i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0283i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.999 - 0.0283i$
Analytic conductor: \(3.74862\)
Root analytic conductor: \(1.93613\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7/2),\ 0.999 - 0.0283i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.65349 + 0.0375668i\)
\(L(\frac12)\) \(\approx\) \(2.65349 + 0.0375668i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10.3 - 4.47i)T \)
3 \( 1 + (-31.1 + 34.8i)T \)
good5 \( 1 + 205. iT - 7.81e4T^{2} \)
7 \( 1 - 999. iT - 8.23e5T^{2} \)
11 \( 1 + 1.97e3T + 1.94e7T^{2} \)
13 \( 1 + 1.27e4T + 6.27e7T^{2} \)
17 \( 1 + 1.88e4iT - 4.10e8T^{2} \)
19 \( 1 - 2.39e4iT - 8.93e8T^{2} \)
23 \( 1 - 7.23e4T + 3.40e9T^{2} \)
29 \( 1 - 1.06e5iT - 1.72e10T^{2} \)
31 \( 1 + 9.03e4iT - 2.75e10T^{2} \)
37 \( 1 - 4.33e4T + 9.49e10T^{2} \)
41 \( 1 + 7.85e5iT - 1.94e11T^{2} \)
43 \( 1 + 7.08e4iT - 2.71e11T^{2} \)
47 \( 1 - 5.63e5T + 5.06e11T^{2} \)
53 \( 1 - 3.48e5iT - 1.17e12T^{2} \)
59 \( 1 + 7.58e5T + 2.48e12T^{2} \)
61 \( 1 - 3.14e5T + 3.14e12T^{2} \)
67 \( 1 + 8.11e5iT - 6.06e12T^{2} \)
71 \( 1 - 2.65e5T + 9.09e12T^{2} \)
73 \( 1 + 2.59e5T + 1.10e13T^{2} \)
79 \( 1 - 5.23e6iT - 1.92e13T^{2} \)
83 \( 1 + 1.01e7T + 2.71e13T^{2} \)
89 \( 1 + 3.88e6iT - 4.42e13T^{2} \)
97 \( 1 - 7.24e6T + 8.07e13T^{2} \)
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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

−18.69073084381251825538070647307, −17.02965915406221938645016409353, −15.44175676989351844538647864383, −14.32332380056691747118073611979, −12.77999850420256112508093841107, −12.08737095135689565051681197017, −8.918592367736754325859666628811, −7.34238542360966431205510754993, −5.26499654883386364413997526171, −2.56429165814071843872503216317, 2.86836281567222283482011616891, 4.62639712078133131638049165746, 7.23475880578486113861851601606, 9.996554052835943321498598308815, 10.95904138341729437493402470583, 13.12695706556131790488637152114, 14.41719547458405387387178130918, 15.22133965879260696055992165568, 16.90357725625264654393604427994, 19.23963543951769138768537913006

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