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Properties

Label 2-12-12.11-c13-0-13
Degree $2$
Conductor $12$
Sign $-0.0752 - 0.997i$
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  + (63.1 + 64.8i)2-s + (1.25e3 − 128. i)3-s + (−218. + 8.18e3i)4-s − 1.78e3i·5-s + (8.76e4 + 7.33e4i)6-s + 3.16e5i·7-s + (−5.44e5 + 5.02e5i)8-s + (1.56e6 − 3.22e5i)9-s + (1.16e5 − 1.12e5i)10-s + 5.24e6·11-s + (7.78e5 + 1.03e7i)12-s − 7.16e6·13-s + (−2.05e7 + 2.00e7i)14-s + (−2.29e5 − 2.24e6i)15-s + (−6.70e7 − 3.57e6i)16-s + 1.30e8i·17-s + ⋯
L(s)  = 1  + (0.697 + 0.716i)2-s + (0.994 − 0.101i)3-s + (−0.0266 + 0.999i)4-s − 0.0511i·5-s + (0.766 + 0.641i)6-s + 1.01i·7-s + (−0.734 + 0.678i)8-s + (0.979 − 0.202i)9-s + (0.0366 − 0.0357i)10-s + 0.892·11-s + (0.0752 + 0.997i)12-s − 0.411·13-s + (−0.729 + 0.710i)14-s + (−0.00521 − 0.0509i)15-s + (−0.998 − 0.0533i)16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0752 - 0.997i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.0752 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.0752 - 0.997i$
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: \((2,\ 12,\ (\ :13/2),\ -0.0752 - 0.997i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.39408 + 2.58153i\)
\(L(\frac12)\) \(\approx\) \(2.39408 + 2.58153i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-63.1 - 64.8i)T \)
3 \( 1 + (-1.25e3 + 128. i)T \)
good5 \( 1 + 1.78e3iT - 1.22e9T^{2} \)
7 \( 1 - 3.16e5iT - 9.68e10T^{2} \)
11 \( 1 - 5.24e6T + 3.45e13T^{2} \)
13 \( 1 + 7.16e6T + 3.02e14T^{2} \)
17 \( 1 - 1.30e8iT - 9.90e15T^{2} \)
19 \( 1 + 1.09e8iT - 4.20e16T^{2} \)
23 \( 1 + 5.41e8T + 5.04e17T^{2} \)
29 \( 1 + 4.73e9iT - 1.02e19T^{2} \)
31 \( 1 + 8.25e9iT - 2.44e19T^{2} \)
37 \( 1 - 1.92e10T + 2.43e20T^{2} \)
41 \( 1 - 4.37e10iT - 9.25e20T^{2} \)
43 \( 1 + 6.99e10iT - 1.71e21T^{2} \)
47 \( 1 - 3.55e10T + 5.46e21T^{2} \)
53 \( 1 + 7.53e10iT - 2.60e22T^{2} \)
59 \( 1 + 2.47e11T + 1.04e23T^{2} \)
61 \( 1 + 5.16e11T + 1.61e23T^{2} \)
67 \( 1 - 1.50e11iT - 5.48e23T^{2} \)
71 \( 1 + 3.92e11T + 1.16e24T^{2} \)
73 \( 1 - 5.99e11T + 1.67e24T^{2} \)
79 \( 1 - 1.31e11iT - 4.66e24T^{2} \)
83 \( 1 + 1.71e12T + 8.87e24T^{2} \)
89 \( 1 - 3.37e12iT - 2.19e25T^{2} \)
97 \( 1 - 1.18e13T + 6.73e25T^{2} \)
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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

−17.01510136056302704635970500031, −15.36315559971416272730011121904, −14.68628033403736438990022733427, −13.24481846162828692732438494423, −12.01542826578186354495444032589, −9.232191889952242834254407028446, −7.994368273061784675581305902604, −6.22716892117006622930246494887, −4.13881212828986084333308760549, −2.40354870030297226712662370634, 1.26401733422526225660808605589, 3.12428064464102411992658028556, 4.50478963149231557091401307315, 7.05660520276972289562398836811, 9.270165790790083935615209379643, 10.58786786399194109379948154698, 12.39253869701371087653146566182, 13.90751456808921783514681524831, 14.50373822046177829490346633220, 16.23343866203691037155861028660

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