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Properties

Label 2-12-4.3-c8-0-1
Degree $2$
Conductor $12$
Sign $-0.257 - 0.966i$
Analytic cond. $4.88854$
Root an. cond. $2.21100$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−15.8 + 2.07i)2-s − 46.7i·3-s + (247. − 65.9i)4-s − 374.·5-s + (97.1 + 741. i)6-s + 4.47e3i·7-s + (−3.78e3 + 1.56e3i)8-s − 2.18e3·9-s + (5.94e3 − 779. i)10-s + 1.29e4i·11-s + (−3.08e3 − 1.15e4i)12-s + 1.75e4·13-s + (−9.29e3 − 7.09e4i)14-s + 1.75e4i·15-s + (5.68e4 − 3.26e4i)16-s − 9.93e4·17-s + ⋯
L(s)  = 1  + (−0.991 + 0.129i)2-s − 0.577i·3-s + (0.966 − 0.257i)4-s − 0.599·5-s + (0.0749 + 0.572i)6-s + 1.86i·7-s + (−0.924 + 0.380i)8-s − 0.333·9-s + (0.594 − 0.0779i)10-s + 0.883i·11-s + (−0.148 − 0.557i)12-s + 0.614·13-s + (−0.241 − 1.84i)14-s + 0.346i·15-s + (0.867 − 0.497i)16-s − 1.18·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.257 - 0.966i$
Analytic conductor: \(4.88854\)
Root analytic conductor: \(2.21100\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :4),\ -0.257 - 0.966i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.360545 + 0.469238i\)
\(L(\frac12)\) \(\approx\) \(0.360545 + 0.469238i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (15.8 - 2.07i)T \)
3 \( 1 + 46.7iT \)
good5 \( 1 + 374.T + 3.90e5T^{2} \)
7 \( 1 - 4.47e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.29e4iT - 2.14e8T^{2} \)
13 \( 1 - 1.75e4T + 8.15e8T^{2} \)
17 \( 1 + 9.93e4T + 6.97e9T^{2} \)
19 \( 1 - 1.15e5iT - 1.69e10T^{2} \)
23 \( 1 - 1.09e4iT - 7.83e10T^{2} \)
29 \( 1 - 3.12e4T + 5.00e11T^{2} \)
31 \( 1 + 1.20e6iT - 8.52e11T^{2} \)
37 \( 1 + 1.24e6T + 3.51e12T^{2} \)
41 \( 1 - 2.96e6T + 7.98e12T^{2} \)
43 \( 1 - 2.69e6iT - 1.16e13T^{2} \)
47 \( 1 - 8.92e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.11e7T + 6.22e13T^{2} \)
59 \( 1 + 6.19e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.13e6T + 1.91e14T^{2} \)
67 \( 1 - 2.55e6iT - 4.06e14T^{2} \)
71 \( 1 + 2.55e7iT - 6.45e14T^{2} \)
73 \( 1 - 3.32e7T + 8.06e14T^{2} \)
79 \( 1 + 2.18e7iT - 1.51e15T^{2} \)
83 \( 1 - 3.75e7iT - 2.25e15T^{2} \)
89 \( 1 - 3.61e7T + 3.93e15T^{2} \)
97 \( 1 + 1.26e7T + 7.83e15T^{2} \)
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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

−18.58314000229630362487791118387, −17.73675697558022172811113302928, −15.87553275849724051157987396118, −15.01486500587175018440414581045, −12.45891071678758272556666840801, −11.41807921713172897505958205779, −9.230704299319521431301217267781, −7.959269109640684145760727078194, −6.10975395011850055348218714446, −2.15961332026874104721778449404, 0.49890304802778352802855119592, 3.77275120628520051869933394327, 7.00699456692170286256039363619, 8.626961538148135148581868801348, 10.46338453126440106213722156546, 11.28594988137064689898246419631, 13.64342918839878059362095254189, 15.62094035294319559486317325308, 16.58289464212890799255540039858, 17.72036833137877016325557160690

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