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Properties

Label 2-84-28.11-c2-0-8
Degree $2$
Conductor $84$
Sign $0.997 + 0.0633i$
Analytic cond. $2.28883$
Root an. cond. $1.51288$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (−1.5 − 0.866i)3-s + 4·4-s + (2.5 + 4.33i)5-s + (−3 − 1.73i)6-s + (3.5 − 6.06i)7-s + 8·8-s + (1.5 + 2.59i)9-s + (5 + 8.66i)10-s + (−4.5 − 2.59i)11-s + (−6 − 3.46i)12-s − 16·13-s + (7 − 12.1i)14-s − 8.66i·15-s + 16·16-s + (−2 + 3.46i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.5 − 0.288i)3-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.288i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.166 + 0.288i)9-s + (0.5 + 0.866i)10-s + (−0.409 − 0.236i)11-s + (−0.5 − 0.288i)12-s − 1.23·13-s + (0.5 − 0.866i)14-s − 0.577i·15-s + 16-s + (−0.117 + 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.997 + 0.0633i$
Analytic conductor: \(2.28883\)
Root analytic conductor: \(1.51288\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1),\ 0.997 + 0.0633i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.06703 - 0.0655212i\)
\(L(\frac12)\) \(\approx\) \(2.06703 - 0.0655212i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-3.5 + 6.06i)T \)
good5 \( 1 + (-2.5 - 4.33i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (4.5 + 2.59i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 16T + 169T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (24 - 13.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (18 - 10.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 37T + 841T^{2} \)
31 \( 1 + (16.5 + 9.52i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (34 + 58.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 40T + 1.68e3T^{2} \)
43 \( 1 - 24.2iT - 1.84e3T^{2} \)
47 \( 1 + (-39 + 22.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (36.5 - 63.2i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-25.5 - 14.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-41 - 71.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-69 - 39.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 72.7iT - 5.04e3T^{2} \)
73 \( 1 + (-47 + 81.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (7.5 - 4.33i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 133. iT - 6.88e3T^{2} \)
89 \( 1 + (29 + 50.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 47T + 9.40e3T^{2} \)
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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

−14.17285045849906698044702080429, −12.99979294616877434467469513364, −12.01225876565518390539590373398, −10.73585960061709763888792712568, −10.29823703221724646655568486885, −7.77979170416004468449810129285, −6.82079267474119289290902028871, −5.68433374736583563773831847083, −4.22824547340236312234561647853, −2.28193495005688848220408935848, 2.28591142774753429370751075064, 4.68366005164014824558757972863, 5.26497384542563367976130865597, 6.63810942251852906878189112545, 8.348104116986700359735553130909, 9.799255701466730236169632131181, 11.03572667436487670750980800223, 12.27613139303076010938705314633, 12.66530501092555230545347599007, 14.03747611337561121178556009283

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