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Properties

Label 2-12-4.3-c16-0-1
Degree $2$
Conductor $12$
Sign $-0.179 - 0.983i$
Analytic cond. $19.4789$
Root an. cond. $4.41349$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−254. + 23.0i)2-s − 3.78e3i·3-s + (6.44e4 − 1.17e4i)4-s − 3.63e5·5-s + (8.71e4 + 9.65e5i)6-s − 2.07e6i·7-s + (−1.61e7 + 4.47e6i)8-s − 1.43e7·9-s + (9.26e7 − 8.36e6i)10-s − 4.00e8i·11-s + (−4.44e7 − 2.44e8i)12-s + 2.75e8·13-s + (4.78e7 + 5.29e8i)14-s + 1.37e9i·15-s + (4.01e9 − 1.51e9i)16-s − 7.49e9·17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0898i)2-s − 0.577i·3-s + (0.983 − 0.179i)4-s − 0.930·5-s + (0.0518 + 0.575i)6-s − 0.360i·7-s + (−0.963 + 0.266i)8-s − 0.333·9-s + (0.926 − 0.0836i)10-s − 1.87i·11-s + (−0.103 − 0.568i)12-s + 0.338·13-s + (0.0323 + 0.358i)14-s + 0.537i·15-s + (0.935 − 0.352i)16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(19.4789\)
Root analytic conductor: \(4.41349\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :8),\ -0.179 - 0.983i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.130970 + 0.156954i\)
\(L(\frac12)\) \(\approx\) \(0.130970 + 0.156954i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (254. - 23.0i)T \)
3 \( 1 + 3.78e3iT \)
good5 \( 1 + 3.63e5T + 1.52e11T^{2} \)
7 \( 1 + 2.07e6iT - 3.32e13T^{2} \)
11 \( 1 + 4.00e8iT - 4.59e16T^{2} \)
13 \( 1 - 2.75e8T + 6.65e17T^{2} \)
17 \( 1 + 7.49e9T + 4.86e19T^{2} \)
19 \( 1 - 1.26e10iT - 2.88e20T^{2} \)
23 \( 1 - 1.11e11iT - 6.13e21T^{2} \)
29 \( 1 - 4.40e11T + 2.50e23T^{2} \)
31 \( 1 - 9.68e11iT - 7.27e23T^{2} \)
37 \( 1 - 2.64e12T + 1.23e25T^{2} \)
41 \( 1 + 7.96e12T + 6.37e25T^{2} \)
43 \( 1 - 6.27e12iT - 1.36e26T^{2} \)
47 \( 1 - 1.17e13iT - 5.66e26T^{2} \)
53 \( 1 + 5.56e13T + 3.87e27T^{2} \)
59 \( 1 - 2.82e14iT - 2.15e28T^{2} \)
61 \( 1 + 1.86e14T + 3.67e28T^{2} \)
67 \( 1 + 8.81e13iT - 1.64e29T^{2} \)
71 \( 1 + 1.09e15iT - 4.16e29T^{2} \)
73 \( 1 - 5.03e14T + 6.50e29T^{2} \)
79 \( 1 + 1.56e15iT - 2.30e30T^{2} \)
83 \( 1 + 8.07e14iT - 5.07e30T^{2} \)
89 \( 1 + 5.95e15T + 1.54e31T^{2} \)
97 \( 1 + 2.50e15T + 6.14e31T^{2} \)
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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

−16.59754541616549168960059284702, −15.59008977685527938941024762248, −13.74314986839639700728384821832, −11.80246743252071610282272591734, −10.84814744515182737769317461926, −8.729128732534755912665493435283, −7.70428100236302529790241218030, −6.17597504628608210679249808366, −3.29542566844191670389385202213, −1.16203593641942243829007164462, 0.11601814435952020806809787462, 2.34588293343594323782995749667, 4.36564717601459210040003668641, 6.85880786492297994560305942598, 8.382213581793013923484914487285, 9.749809200171698632711754348244, 11.17167176258379054979178452350, 12.39667720459369252224939254302, 15.13422145469981429515260859530, 15.68314224531009380315439882239

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