Properties

Label 2-84-28.27-c11-0-20
Degree $2$
Conductor $84$
Sign $0.604 + 0.796i$
Analytic cond. $64.5408$
Root an. cond. $8.03373$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (19.2 − 40.9i)2-s − 243·3-s + (−1.30e3 − 1.57e3i)4-s − 4.16e3i·5-s + (−4.66e3 + 9.95e3i)6-s + (−4.44e4 − 1.96e3i)7-s + (−8.96e4 + 2.33e4i)8-s + 5.90e4·9-s + (−1.70e5 − 8.00e4i)10-s + 1.98e5i·11-s + (3.18e5 + 3.82e5i)12-s + 1.32e6i·13-s + (−9.34e5 + 1.78e6i)14-s + 1.01e6i·15-s + (−7.64e5 + 4.12e6i)16-s − 5.45e6i·17-s + ⋯
L(s)  = 1  + (0.424 − 0.905i)2-s − 0.577·3-s + (−0.639 − 0.768i)4-s − 0.596i·5-s + (−0.245 + 0.522i)6-s + (−0.999 − 0.0442i)7-s + (−0.967 + 0.252i)8-s + 0.333·9-s + (−0.539 − 0.253i)10-s + 0.371i·11-s + (0.369 + 0.443i)12-s + 0.990i·13-s + (−0.464 + 0.885i)14-s + 0.344i·15-s + (−0.182 + 0.983i)16-s − 0.931i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ 0.604 + 0.796i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.01902 - 0.505708i\)
\(L(\frac12)\) \(\approx\) \(1.01902 - 0.505708i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-19.2 + 40.9i)T \)
3 \( 1 + 243T \)
7 \( 1 + (4.44e4 + 1.96e3i)T \)
good5 \( 1 + 4.16e3iT - 4.88e7T^{2} \)
11 \( 1 - 1.98e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.32e6iT - 1.79e12T^{2} \)
17 \( 1 + 5.45e6iT - 3.42e13T^{2} \)
19 \( 1 + 2.12e6T + 1.16e14T^{2} \)
23 \( 1 - 3.38e7iT - 9.52e14T^{2} \)
29 \( 1 + 7.98e7T + 1.22e16T^{2} \)
31 \( 1 + 2.62e8T + 2.54e16T^{2} \)
37 \( 1 - 5.78e8T + 1.77e17T^{2} \)
41 \( 1 - 2.33e8iT - 5.50e17T^{2} \)
43 \( 1 + 8.25e8iT - 9.29e17T^{2} \)
47 \( 1 + 5.06e8T + 2.47e18T^{2} \)
53 \( 1 - 1.49e9T + 9.26e18T^{2} \)
59 \( 1 - 3.25e9T + 3.01e19T^{2} \)
61 \( 1 + 6.97e9iT - 4.35e19T^{2} \)
67 \( 1 - 3.10e9iT - 1.22e20T^{2} \)
71 \( 1 + 7.71e9iT - 2.31e20T^{2} \)
73 \( 1 + 1.09e10iT - 3.13e20T^{2} \)
79 \( 1 + 4.30e10iT - 7.47e20T^{2} \)
83 \( 1 + 2.39e10T + 1.28e21T^{2} \)
89 \( 1 - 7.05e10iT - 2.77e21T^{2} \)
97 \( 1 - 9.93e10iT - 7.15e21T^{2} \)
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

−11.91915597909606452929756258979, −11.01045019187446009102766829606, −9.673752499637992149069680382682, −9.134355182003784934630031253672, −7.07314561550626688812075065544, −5.77971621080595648428440989789, −4.69252882041772564154908710289, −3.53362480304126829386509625067, −1.97674376023546872028666744252, −0.65901842875203459377210414666, 0.43379852615505558739940632252, 2.87634324235398295977136117792, 3.99593806711284443981357651771, 5.58256546084578198614791895073, 6.33039909311706514598755343520, 7.33335099284452965011957650698, 8.650149813795085686041905554071, 10.01295460509803963285140794648, 11.10006706475988825681638392092, 12.69751168680088911791446319152

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