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Properties

Label 2-12-3.2-c16-0-1
Degree $2$
Conductor $12$
Sign $-0.969 + 0.243i$
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  + (−6.36e3 + 1.59e3i)3-s + 7.46e5i·5-s + 4.25e6·7-s + (3.79e7 − 2.03e7i)9-s + 2.12e8i·11-s − 7.06e8·13-s + (−1.19e9 − 4.75e9i)15-s + 8.78e9i·17-s − 8.43e9·19-s + (−2.70e10 + 6.79e9i)21-s − 1.19e11i·23-s − 4.05e11·25-s + (−2.08e11 + 1.90e11i)27-s − 3.07e11i·29-s + 2.63e10·31-s + ⋯
L(s)  = 1  + (−0.969 + 0.243i)3-s + 1.91i·5-s + 0.737·7-s + (0.881 − 0.472i)9-s + 0.992i·11-s − 0.865·13-s + (−0.465 − 1.85i)15-s + 1.25i·17-s − 0.496·19-s + (−0.715 + 0.179i)21-s − 1.53i·23-s − 2.65·25-s + (−0.739 + 0.672i)27-s − 0.615i·29-s + 0.0308·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.969 + 0.243i$
Analytic conductor: \(19.4789\)
Root analytic conductor: \(4.41349\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :8),\ -0.969 + 0.243i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.0919515 - 0.743649i\)
\(L(\frac12)\) \(\approx\) \(0.0919515 - 0.743649i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (6.36e3 - 1.59e3i)T \)
good5 \( 1 - 7.46e5iT - 1.52e11T^{2} \)
7 \( 1 - 4.25e6T + 3.32e13T^{2} \)
11 \( 1 - 2.12e8iT - 4.59e16T^{2} \)
13 \( 1 + 7.06e8T + 6.65e17T^{2} \)
17 \( 1 - 8.78e9iT - 4.86e19T^{2} \)
19 \( 1 + 8.43e9T + 2.88e20T^{2} \)
23 \( 1 + 1.19e11iT - 6.13e21T^{2} \)
29 \( 1 + 3.07e11iT - 2.50e23T^{2} \)
31 \( 1 - 2.63e10T + 7.27e23T^{2} \)
37 \( 1 + 3.10e11T + 1.23e25T^{2} \)
41 \( 1 + 5.28e12iT - 6.37e25T^{2} \)
43 \( 1 - 1.92e13T + 1.36e26T^{2} \)
47 \( 1 - 2.19e12iT - 5.66e26T^{2} \)
53 \( 1 + 1.73e13iT - 3.87e27T^{2} \)
59 \( 1 - 1.68e14iT - 2.15e28T^{2} \)
61 \( 1 + 2.49e14T + 3.67e28T^{2} \)
67 \( 1 + 1.53e14T + 1.64e29T^{2} \)
71 \( 1 - 1.82e14iT - 4.16e29T^{2} \)
73 \( 1 + 4.12e14T + 6.50e29T^{2} \)
79 \( 1 - 1.16e15T + 2.30e30T^{2} \)
83 \( 1 - 2.11e15iT - 5.07e30T^{2} \)
89 \( 1 + 5.45e15iT - 1.54e31T^{2} \)
97 \( 1 + 9.42e15T + 6.14e31T^{2} \)
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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

−17.23886181369635192485796716956, −15.21547073624607922349269872654, −14.55881033053428350026126288930, −12.30114235582870064193151824506, −10.91033821211018912871389208438, −10.16893159440793796773345629121, −7.40350867043986406072618456250, −6.22964212069033499464745795887, −4.29780165744016960879897886365, −2.20415032095252508291977939000, 0.32081524903498784983758548370, 1.43479365768162528180342393087, 4.65810639867388815092374710489, 5.52602467919457870438717818383, 7.77969660179754618359833400286, 9.319402808242157274298555545634, 11.36851562292727053994028799883, 12.38092251721930783204842867058, 13.62371000024914499818440360404, 15.91757606790748614571389331098

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