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Properties

Label 2-12-12.11-c13-0-16
Degree $2$
Conductor $12$
Sign $0.876 + 0.481i$
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  + (83.2 + 35.5i)2-s + (−1.20e3 + 377. i)3-s + (5.66e3 + 5.91e3i)4-s − 5.35e4i·5-s + (−1.13e5 − 1.13e4i)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 + (−9.06e6 − 4.98e6i)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 + (−0.995 − 0.0992i)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 + (−0.876 − 0.481i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.876 + 0.481i$
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: \((2,\ 12,\ (\ :13/2),\ 0.876 + 0.481i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.30537 - 0.591973i\)
\(L(\frac12)\) \(\approx\) \(2.30537 - 0.591973i\)
\(L(\frac{15}{2})\) 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 \( 1 + 2.26e12T + 6.73e25T^{2} \)
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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

−16.70222529635645956150929853545, −15.69302052769099434711866685352, −13.80988513448238504276504713856, −12.40379643926574881276839560470, −11.50968300752852457766722475636, −9.178646759103263047061223380741, −6.91623483920207293582106065999, −5.21599007514384009719724890925, −4.20137237929370956004041135225, −0.942992720546992405897516585481, 1.74993284487411516736085167078, 3.75896488688293721953476722234, 5.96639032066150006082589283966, 6.87051782880717366279928743687, 10.29210462061958450537906821714, 11.34898506413480685807593821383, 12.45526452683035937065311101593, 14.24164274591507552615535771810, 15.21954242143276548375045938218, 17.04421736513041639824769467720

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