↑

Properties

Label 2-84-12.11-c1-0-1
Degree $2$
Conductor $84$
Sign $-0.579 - 0.815i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.349 + 1.37i)2-s + (−1.72 + 0.203i)3-s + (−1.75 + 0.958i)4-s + 2.27i·5-s + (−0.880 − 2.28i)6-s + i·7-s + (−1.92 − 2.06i)8-s + (2.91 − 0.699i)9-s + (−3.12 + 0.796i)10-s + 4.31·11-s + (2.82 − 2.00i)12-s − 0.406·13-s + (−1.37 + 0.349i)14-s + (−0.463 − 3.91i)15-s + (2.16 − 3.36i)16-s − 4.31i·17-s + ⋯
L(s)  = 1  + (0.247 + 0.968i)2-s + (−0.993 + 0.117i)3-s + (−0.877 + 0.479i)4-s + 1.01i·5-s + (−0.359 − 0.933i)6-s + 0.377i·7-s + (−0.681 − 0.731i)8-s + (0.972 − 0.233i)9-s + (−0.986 + 0.251i)10-s + 1.30·11-s + (0.815 − 0.579i)12-s − 0.112·13-s + (−0.366 + 0.0934i)14-s + (−0.119 − 1.01i)15-s + (0.540 − 0.841i)16-s − 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.579 - 0.815i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ -0.579 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355215 + 0.687979i\)
\(L(\frac12)\) \(\approx\) \(0.355215 + 0.687979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.349 - 1.37i)T \)
3 \( 1 + (1.72 - 0.203i)T \)
7 \( 1 - iT \)
good5 \( 1 - 2.27iT - 5T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 + 0.406T + 13T^{2} \)
17 \( 1 + 4.31iT - 17T^{2} \)
19 \( 1 - 5.42iT - 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 - 3.72iT - 29T^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + 7.83T + 37T^{2} \)
41 \( 1 + 2.56iT - 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 - 2.32T + 47T^{2} \)
53 \( 1 - 5.48iT - 53T^{2} \)
59 \( 1 - 3.91T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 7.83iT - 67T^{2} \)
71 \( 1 + 4.88T + 71T^{2} \)
73 \( 1 - 2.81T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 0.641T + 83T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−14.72991515149737289663560296749, −13.92542886168324957774443654594, −12.40727709113359965946896500485, −11.62531553815766633727238088397, −10.24316535123926160564823581734, −9.069501378673835017535979028270, −7.28633647865419757907446770569, −6.51656590076277413526604915996, −5.39540967448921631631149239044, −3.79170952090372095817767301368, 1.23268990940303798440357755413, 4.10845721088601200187235172368, 5.13072754573617094397881489375, 6.59647803541515862415146053023, 8.610006673338277973169977923694, 9.707625398429553857920881174750, 10.90189492122606933178466181900, 11.81320311044038914048390802957, 12.65252900996271355841378805107, 13.42698639863552038067355066484

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