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Properties

Label 2-12-3.2-c18-0-1
Degree $2$
Conductor $12$
Sign $0.727 - 0.685i$
Analytic cond. $24.6463$
Root an. cond. $4.96450$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.43e4 + 1.34e4i)3-s − 4.80e5i·5-s − 7.51e6·7-s + (2.31e7 − 3.86e8i)9-s − 2.15e9i·11-s − 1.26e9·13-s + (6.48e9 + 6.88e9i)15-s + 1.24e11i·17-s + 1.23e11·19-s + (1.07e11 − 1.01e11i)21-s + 1.00e11i·23-s + 3.58e12·25-s + (4.88e12 + 5.85e12i)27-s + 2.73e13i·29-s + 3.40e13·31-s + ⋯
L(s)  = 1  + (−0.727 + 0.685i)3-s − 0.246i·5-s − 0.186·7-s + (0.0597 − 0.998i)9-s − 0.915i·11-s − 0.119·13-s + (0.168 + 0.179i)15-s + 1.05i·17-s + 0.383·19-s + (0.135 − 0.127i)21-s + 0.0560i·23-s + 0.939·25-s + (0.640 + 0.767i)27-s + 1.88i·29-s + 1.28·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.727 - 0.685i$
Analytic conductor: \(24.6463\)
Root analytic conductor: \(4.96450\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :9),\ 0.727 - 0.685i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.337762062\)
\(L(\frac12)\) \(\approx\) \(1.337762062\)
\(L(10)\) 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 \( 1 + (1.43e4 - 1.34e4i)T \)
good5 \( 1 + 4.80e5iT - 3.81e12T^{2} \)
7 \( 1 + 7.51e6T + 1.62e15T^{2} \)
11 \( 1 + 2.15e9iT - 5.55e18T^{2} \)
13 \( 1 + 1.26e9T + 1.12e20T^{2} \)
17 \( 1 - 1.24e11iT - 1.40e22T^{2} \)
19 \( 1 - 1.23e11T + 1.04e23T^{2} \)
23 \( 1 - 1.00e11iT - 3.24e24T^{2} \)
29 \( 1 - 2.73e13iT - 2.10e26T^{2} \)
31 \( 1 - 3.40e13T + 6.99e26T^{2} \)
37 \( 1 - 1.11e14T + 1.68e28T^{2} \)
41 \( 1 - 3.61e14iT - 1.07e29T^{2} \)
43 \( 1 + 1.64e14T + 2.52e29T^{2} \)
47 \( 1 + 1.71e15iT - 1.25e30T^{2} \)
53 \( 1 + 2.70e15iT - 1.08e31T^{2} \)
59 \( 1 - 6.75e15iT - 7.50e31T^{2} \)
61 \( 1 - 7.85e15T + 1.36e32T^{2} \)
67 \( 1 - 3.25e16T + 7.40e32T^{2} \)
71 \( 1 - 4.47e16iT - 2.10e33T^{2} \)
73 \( 1 - 7.24e16T + 3.46e33T^{2} \)
79 \( 1 - 2.22e16T + 1.43e34T^{2} \)
83 \( 1 + 2.78e17iT - 3.49e34T^{2} \)
89 \( 1 - 6.88e15iT - 1.22e35T^{2} \)
97 \( 1 - 1.33e18T + 5.77e35T^{2} \)
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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

−16.08244623959649606720306047247, −14.66807169825780722974819739186, −12.85896756278567840902066452724, −11.42231837269590145547744919172, −10.17519312385143122418085519265, −8.636474273217976319326825061078, −6.43173920076587753336401914773, −5.03494589168739918179038446549, −3.41459318496762096937772569327, −0.915192874751371619902568423125, 0.70097503350911959696629463528, 2.43938439935853465915442663729, 4.79393443911367223292649875916, 6.44470376444093978883204694362, 7.67335826336287834846462444409, 9.825761564912211119399744236161, 11.39697840328207400278757318616, 12.56343537334128393653812122666, 13.92807119471990712042709685207, 15.65296929383083420864467072529

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