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Properties

Label 2-84-12.11-c5-0-19
Degree $2$
Conductor $84$
Sign $-0.765 - 0.643i$
Analytic cond. $13.4722$
Root an. cond. $3.67045$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.65 + 0.276i)2-s + (−8.82 + 12.8i)3-s + (31.8 + 3.12i)4-s + 100. i·5-s + (−53.4 + 70.1i)6-s − 49i·7-s + (179. + 26.4i)8-s + (−87.2 − 226. i)9-s + (−27.6 + 565. i)10-s − 5.60·11-s + (−321. + 381. i)12-s − 1.04e3·13-s + (13.5 − 276. i)14-s + (−1.28e3 − 883. i)15-s + (1.00e3 + 198. i)16-s + 333. i·17-s + ⋯
L(s)  = 1  + (0.998 + 0.0488i)2-s + (−0.566 + 0.824i)3-s + (0.995 + 0.0975i)4-s + 1.79i·5-s + (−0.605 + 0.795i)6-s − 0.377i·7-s + (0.989 + 0.146i)8-s + (−0.358 − 0.933i)9-s + (−0.0874 + 1.78i)10-s − 0.0139·11-s + (−0.643 + 0.765i)12-s − 1.71·13-s + (0.0184 − 0.377i)14-s + (−1.47 − 1.01i)15-s + (0.980 + 0.194i)16-s + 0.279i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.765 - 0.643i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ -0.765 - 0.643i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.811127 + 2.22348i\)
\(L(\frac12)\) \(\approx\) \(0.811127 + 2.22348i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 - 0.276i)T \)
3 \( 1 + (8.82 - 12.8i)T \)
7 \( 1 + 49iT \)
good5 \( 1 - 100. iT - 3.12e3T^{2} \)
11 \( 1 + 5.60T + 1.61e5T^{2} \)
13 \( 1 + 1.04e3T + 3.71e5T^{2} \)
17 \( 1 - 333. iT - 1.41e6T^{2} \)
19 \( 1 - 1.63e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.80e3T + 6.43e6T^{2} \)
29 \( 1 - 3.37e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.26e3iT - 2.86e7T^{2} \)
37 \( 1 - 2.88e3T + 6.93e7T^{2} \)
41 \( 1 - 680. iT - 1.15e8T^{2} \)
43 \( 1 + 1.09e4iT - 1.47e8T^{2} \)
47 \( 1 + 3.23e3T + 2.29e8T^{2} \)
53 \( 1 - 3.16e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.68e4T + 7.14e8T^{2} \)
61 \( 1 - 2.47e4T + 8.44e8T^{2} \)
67 \( 1 - 3.14e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.66e4T + 1.80e9T^{2} \)
73 \( 1 - 6.91e4T + 2.07e9T^{2} \)
79 \( 1 - 2.56e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.31e4T + 3.93e9T^{2} \)
89 \( 1 + 7.83e4iT - 5.58e9T^{2} \)
97 \( 1 + 8.62e4T + 8.58e9T^{2} \)
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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

−14.13476023423403372380300171587, −12.51932117857791636037411104588, −11.44460037548220828425813433136, −10.62781214387047819660444326845, −9.917501723248082571595986378083, −7.41346617852586734671390349794, −6.59102999386025911627533183282, −5.29323329409033052942833429554, −3.84493678493584150235908436746, −2.70511662558157729263095340783, 0.74953769158305675442226443545, 2.32050001275252202408379434117, 4.85605152463149893673805974437, 5.22034295946761928450006174397, 6.82561116903121002234755841981, 8.040388689160035710476242156758, 9.516382776297569531509861885921, 11.32360292601377780723470697703, 12.17035102400941542773168846627, 12.80791590068433101417555658404

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