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Properties

Label 2-12-4.3-c16-0-9
Degree $2$
Conductor $12$
Sign $0.981 + 0.192i$
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  + (197. − 162. i)2-s + 3.78e3i·3-s + (1.26e4 − 6.43e4i)4-s + 5.51e5·5-s + (6.16e5 + 7.48e5i)6-s + 6.23e6i·7-s + (−7.95e6 − 1.47e7i)8-s − 1.43e7·9-s + (1.09e8 − 8.97e7i)10-s + 1.63e8i·11-s + (2.43e8 + 4.78e7i)12-s + 6.83e8·13-s + (1.01e9 + 1.23e9i)14-s + 2.09e9i·15-s + (−3.97e9 − 1.62e9i)16-s + 1.25e10·17-s + ⋯
L(s)  = 1  + (0.772 − 0.635i)2-s + 0.577i·3-s + (0.192 − 0.981i)4-s + 1.41·5-s + (0.366 + 0.445i)6-s + 1.08i·7-s + (−0.474 − 0.880i)8-s − 0.333·9-s + (1.09 − 0.897i)10-s + 0.763i·11-s + (0.566 + 0.111i)12-s + 0.837·13-s + (0.687 + 0.835i)14-s + 0.815i·15-s + (−0.925 − 0.378i)16-s + 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.981 + 0.192i$
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: \((2,\ 12,\ (\ :8),\ 0.981 + 0.192i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(3.76156 - 0.366230i\)
\(L(\frac12)\) \(\approx\) \(3.76156 - 0.366230i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-197. + 162. i)T \)
3 \( 1 - 3.78e3iT \)
good5 \( 1 - 5.51e5T + 1.52e11T^{2} \)
7 \( 1 - 6.23e6iT - 3.32e13T^{2} \)
11 \( 1 - 1.63e8iT - 4.59e16T^{2} \)
13 \( 1 - 6.83e8T + 6.65e17T^{2} \)
17 \( 1 - 1.25e10T + 4.86e19T^{2} \)
19 \( 1 + 1.86e10iT - 2.88e20T^{2} \)
23 \( 1 - 6.04e10iT - 6.13e21T^{2} \)
29 \( 1 - 5.51e11T + 2.50e23T^{2} \)
31 \( 1 + 8.13e11iT - 7.27e23T^{2} \)
37 \( 1 + 3.30e12T + 1.23e25T^{2} \)
41 \( 1 + 1.01e13T + 6.37e25T^{2} \)
43 \( 1 - 8.88e12iT - 1.36e26T^{2} \)
47 \( 1 - 4.48e12iT - 5.66e26T^{2} \)
53 \( 1 + 1.01e14T + 3.87e27T^{2} \)
59 \( 1 + 5.21e13iT - 2.15e28T^{2} \)
61 \( 1 + 3.28e13T + 3.67e28T^{2} \)
67 \( 1 - 7.59e14iT - 1.64e29T^{2} \)
71 \( 1 + 7.87e14iT - 4.16e29T^{2} \)
73 \( 1 + 7.67e14T + 6.50e29T^{2} \)
79 \( 1 + 2.52e15iT - 2.30e30T^{2} \)
83 \( 1 + 2.72e15iT - 5.07e30T^{2} \)
89 \( 1 - 5.91e14T + 1.54e31T^{2} \)
97 \( 1 - 6.86e14T + 6.14e31T^{2} \)
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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

−15.73054922134153861182203798660, −14.52931945392962287446611808276, −13.30594336454555788921332553702, −11.88363979078797448077533336194, −10.20747789233356831052909795475, −9.265880093978159857542368625106, −6.07132267550133923975124115050, −5.08324108528313672180028897283, −2.98786979893338980473056719325, −1.60511341059975435931289263189, 1.34278651652105894491583883023, 3.34888046106609995850547567129, 5.52749492838414553329624117595, 6.61567704475075765982734966628, 8.257898618148722318561978819899, 10.40140834028883592944480441590, 12.41269496676819815107261897174, 13.85151273741513992660050112154, 14.08071225165578953039770318105, 16.43033052763209152497211020655

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