Properties

Label 2-12-4.3-c16-0-2
Degree $2$
Conductor $12$
Sign $0.0887 - 0.996i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (11.3 − 255. i)2-s + 3.78e3i·3-s + (−6.52e4 − 5.81e3i)4-s + 8.65e4·5-s + (9.68e5 + 4.30e4i)6-s − 1.07e7i·7-s + (−2.23e6 + 1.66e7i)8-s − 1.43e7·9-s + (9.84e5 − 2.21e7i)10-s + 3.08e8i·11-s + (2.20e7 − 2.47e8i)12-s − 9.27e8·13-s + (−2.75e9 − 1.22e8i)14-s + 3.28e8i·15-s + (4.22e9 + 7.59e8i)16-s + 1.04e9·17-s + ⋯
L(s)  = 1  + (0.0444 − 0.999i)2-s + 0.577i·3-s + (−0.996 − 0.0887i)4-s + 0.221·5-s + (0.576 + 0.0256i)6-s − 1.87i·7-s + (−0.132 + 0.991i)8-s − 0.333·9-s + (0.00984 − 0.221i)10-s + 1.44i·11-s + (0.0512 − 0.575i)12-s − 1.13·13-s + (−1.87 − 0.0831i)14-s + 0.127i·15-s + (0.984 + 0.176i)16-s + 0.150·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0887 - 0.996i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.0887 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.0887 - 0.996i$
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: \((2,\ 12,\ (\ :8),\ 0.0887 - 0.996i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.389860 + 0.356661i\)
\(L(\frac12)\) \(\approx\) \(0.389860 + 0.356661i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.3 + 255. i)T \)
3 \( 1 - 3.78e3iT \)
good5 \( 1 - 8.65e4T + 1.52e11T^{2} \)
7 \( 1 + 1.07e7iT - 3.32e13T^{2} \)
11 \( 1 - 3.08e8iT - 4.59e16T^{2} \)
13 \( 1 + 9.27e8T + 6.65e17T^{2} \)
17 \( 1 - 1.04e9T + 4.86e19T^{2} \)
19 \( 1 - 2.14e10iT - 2.88e20T^{2} \)
23 \( 1 - 8.03e10iT - 6.13e21T^{2} \)
29 \( 1 - 6.46e10T + 2.50e23T^{2} \)
31 \( 1 - 1.00e12iT - 7.27e23T^{2} \)
37 \( 1 - 1.20e12T + 1.23e25T^{2} \)
41 \( 1 + 1.21e13T + 6.37e25T^{2} \)
43 \( 1 - 2.28e12iT - 1.36e26T^{2} \)
47 \( 1 + 2.13e13iT - 5.66e26T^{2} \)
53 \( 1 - 1.37e13T + 3.87e27T^{2} \)
59 \( 1 - 1.78e14iT - 2.15e28T^{2} \)
61 \( 1 + 5.51e13T + 3.67e28T^{2} \)
67 \( 1 + 2.57e14iT - 1.64e29T^{2} \)
71 \( 1 + 4.05e14iT - 4.16e29T^{2} \)
73 \( 1 + 2.94e14T + 6.50e29T^{2} \)
79 \( 1 + 1.81e14iT - 2.30e30T^{2} \)
83 \( 1 + 1.39e15iT - 5.07e30T^{2} \)
89 \( 1 - 3.78e15T + 1.54e31T^{2} \)
97 \( 1 - 8.34e15T + 6.14e31T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.88482827940392344510884810898, −14.71301569554935561182273023734, −13.59241023289005917764116151880, −12.06424421571895998356152206873, −10.31215016008534348146972957190, −9.834502662802390696392642968822, −7.52536311611182124616581568155, −4.82486058899885652602260074724, −3.68252727336253859935274688579, −1.62514902593241156704497031745, 0.18863335643725837725540176627, 2.62902984121383472971299881439, 5.28315387854138556619076382648, 6.34598235995845634181195765335, 8.187924361042442342320611272592, 9.269439286924342758971870866721, 11.85050434436326557684097140097, 13.17986082092924450205229885902, 14.58653369491768269259099732663, 15.73920049365050925829825935155

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