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Properties

Label 2-12-4.3-c20-0-7
Degree $2$
Conductor $12$
Sign $0.249 + 0.968i$
Analytic cond. $30.4216$
Root an. cond. $5.51558$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−128. − 1.01e3i)2-s − 3.40e4i·3-s + (−1.01e6 + 2.61e5i)4-s − 1.38e7·5-s + (−3.46e7 + 4.38e6i)6-s + 4.88e8i·7-s + (3.95e8 + 9.98e8i)8-s − 1.16e9·9-s + (1.77e9 + 1.40e10i)10-s − 4.21e10i·11-s + (8.90e9 + 3.46e10i)12-s + 1.60e10·13-s + (4.95e11 − 6.27e10i)14-s + 4.70e11i·15-s + (9.63e11 − 5.30e11i)16-s − 1.25e12·17-s + ⋯
L(s)  = 1  + (−0.125 − 0.992i)2-s − 0.577i·3-s + (−0.968 + 0.249i)4-s − 1.41·5-s + (−0.572 + 0.0724i)6-s + 1.72i·7-s + (0.368 + 0.929i)8-s − 0.333·9-s + (0.177 + 1.40i)10-s − 1.62i·11-s + (0.143 + 0.559i)12-s + 0.116·13-s + (1.71 − 0.216i)14-s + 0.816i·15-s + (0.875 − 0.482i)16-s − 0.622·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(30.4216\)
Root analytic conductor: \(5.51558\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :10),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.8869062490\)
\(L(\frac12)\) \(\approx\) \(0.8869062490\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (128. + 1.01e3i)T \)
3 \( 1 + 3.40e4iT \)
good5 \( 1 + 1.38e7T + 9.53e13T^{2} \)
7 \( 1 - 4.88e8iT - 7.97e16T^{2} \)
11 \( 1 + 4.21e10iT - 6.72e20T^{2} \)
13 \( 1 - 1.60e10T + 1.90e22T^{2} \)
17 \( 1 + 1.25e12T + 4.06e24T^{2} \)
19 \( 1 - 4.07e12iT - 3.75e25T^{2} \)
23 \( 1 + 2.02e12iT - 1.71e27T^{2} \)
29 \( 1 + 4.84e14T + 1.76e29T^{2} \)
31 \( 1 - 1.60e13iT - 6.71e29T^{2} \)
37 \( 1 - 8.83e15T + 2.31e31T^{2} \)
41 \( 1 - 9.52e15T + 1.80e32T^{2} \)
43 \( 1 - 2.56e14iT - 4.67e32T^{2} \)
47 \( 1 + 8.84e16iT - 2.76e33T^{2} \)
53 \( 1 - 2.40e17T + 3.05e34T^{2} \)
59 \( 1 - 1.81e17iT - 2.61e35T^{2} \)
61 \( 1 - 1.16e18T + 5.08e35T^{2} \)
67 \( 1 - 2.69e17iT - 3.32e36T^{2} \)
71 \( 1 - 1.97e18iT - 1.05e37T^{2} \)
73 \( 1 - 2.82e18T + 1.84e37T^{2} \)
79 \( 1 + 1.10e19iT - 8.96e37T^{2} \)
83 \( 1 - 8.89e18iT - 2.40e38T^{2} \)
89 \( 1 - 8.01e18T + 9.72e38T^{2} \)
97 \( 1 + 2.29e19T + 5.43e39T^{2} \)
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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

−14.84718018939439980004726766128, −13.07962887149157026468574053539, −11.85093106922787673539398600675, −11.27518011190409203939751131405, −8.877481094263716221410382464912, −8.091119568192694114930308947458, −5.67519657205242453793570082552, −3.68184342500353719243792250666, −2.41435748516348603395366518761, −0.59043540357695076727356667108, 0.59879985056942844441325544501, 4.00344190072290102338148115175, 4.52047782738626717223935000653, 7.02264048517058692943720357598, 7.79146238154733261548522192042, 9.613890887562905184538985024916, 11.02515138712762018858093395224, 13.02544911357778680671834880264, 14.63188566277242496607630657678, 15.56986650880517407448519683481

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