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Properties

Label 2-84-21.2-c2-0-3
Degree $2$
Conductor $84$
Sign $0.994 + 0.103i$
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  + (2.81 − 1.04i)3-s + (5.12 + 2.95i)5-s − 7·7-s + (6.81 − 5.88i)9-s + (5.12 − 2.95i)11-s − 6·13-s + (17.5 + 2.95i)15-s + (5.12 − 2.95i)17-s + (−11.5 + 19.9i)19-s + (−19.6 + 7.32i)21-s + (−35.8 − 20.7i)23-s + (4.99 + 8.66i)25-s + (13 − 23.6i)27-s + 47.3i·29-s + (−19.5 − 33.7i)31-s + ⋯
L(s)  = 1  + (0.937 − 0.348i)3-s + (1.02 + 0.591i)5-s − 7-s + (0.756 − 0.653i)9-s + (0.465 − 0.268i)11-s − 0.461·13-s + (1.16 + 0.197i)15-s + (0.301 − 0.174i)17-s + (−0.605 + 1.04i)19-s + (−0.937 + 0.348i)21-s + (−1.55 − 0.900i)23-s + (0.199 + 0.346i)25-s + (0.481 − 0.876i)27-s + 1.63i·29-s + (−0.629 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(2.28883\)
Root analytic conductor: \(1.51288\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1),\ 0.994 + 0.103i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.75018 - 0.0911557i\)
\(L(\frac12)\) \(\approx\) \(1.75018 - 0.0911557i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.81 + 1.04i)T \)
7 \( 1 + 7T \)
good5 \( 1 + (-5.12 - 2.95i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.12 + 2.95i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 6T + 169T^{2} \)
17 \( 1 + (-5.12 + 2.95i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.5 - 19.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (35.8 + 20.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 47.3iT - 841T^{2} \)
31 \( 1 + (19.5 + 33.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (23.5 - 40.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 22T + 1.84e3T^{2} \)
47 \( 1 + (-46.1 - 26.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-46.1 + 26.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-87.0 + 50.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-40.5 + 70.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.5 + 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 94.6iT - 5.04e3T^{2} \)
73 \( 1 + (-8.5 - 14.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 47.3iT - 6.88e3T^{2} \)
89 \( 1 + (76.8 + 44.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 82T + 9.40e3T^{2} \)
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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

−14.16107176881054526036119561832, −13.08454220941968421570852962436, −12.19693155252711619214034924946, −10.26398647081592533605251114827, −9.696635757562034646533810799278, −8.448502776988039649902289069227, −6.94031553952475423640789628896, −6.03935264086048908780773257976, −3.63674745671601603537381108088, −2.20506847599306012807319550420, 2.21397067448074450220664230512, 3.98426084438199529642619115358, 5.65463463007371042476101486094, 7.16067018474621325406995579840, 8.754172581138015683801869459687, 9.567414404086492036636754850004, 10.25264281358182712429999735993, 12.18858930404397871577166789660, 13.26732110324494465738244039786, 13.84020114017351617093324331210

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