Properties

Label 2-84-21.2-c8-0-10
Degree $2$
Conductor $84$
Sign $0.511 + 0.859i$
Analytic cond. $34.2198$
Root an. cond. $5.84976$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−40.5 − 70.1i)3-s + (−2.13e3 + 1.09e3i)7-s + (−3.28e3 + 5.68e3i)9-s + 5.64e4·13-s + (−7.89e4 + 1.36e5i)19-s + (1.63e5 + 1.05e5i)21-s + (−1.95e5 − 3.38e5i)25-s + 5.31e5·27-s + (−2.91e5 − 5.05e5i)31-s + (1.73e6 − 3.00e6i)37-s + (−2.28e6 − 3.95e6i)39-s + 3.34e6·43-s + (3.36e6 − 4.68e6i)49-s + 1.27e7·57-s + (1.19e7 − 2.06e7i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.889 + 0.456i)7-s + (−0.5 + 0.866i)9-s + 1.97·13-s + (−0.606 + 1.04i)19-s + (0.840 + 0.542i)21-s + (−0.5 − 0.866i)25-s + 27-s + (−0.315 − 0.547i)31-s + (0.925 − 1.60i)37-s + (−0.988 − 1.71i)39-s + 0.978·43-s + (0.583 − 0.812i)49-s + 1.21·57-s + (0.860 − 1.49i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.511 + 0.859i$
Analytic conductor: \(34.2198\)
Root analytic conductor: \(5.84976\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :4),\ 0.511 + 0.859i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.18314 - 0.672418i\)
\(L(\frac12)\) \(\approx\) \(1.18314 - 0.672418i\)
\(L(5)\) 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 \( 1 + (40.5 + 70.1i)T \)
7 \( 1 + (2.13e3 - 1.09e3i)T \)
good5 \( 1 + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 5.64e4T + 8.15e8T^{2} \)
17 \( 1 + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (7.89e4 - 1.36e5i)T + (-8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 5.00e11T^{2} \)
31 \( 1 + (2.91e5 + 5.05e5i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-1.73e6 + 3.00e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 7.98e12T^{2} \)
43 \( 1 - 3.34e6T + 1.16e13T^{2} \)
47 \( 1 + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.19e7 + 2.06e7i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-1.59e7 - 2.76e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 6.45e14T^{2} \)
73 \( 1 + (1.95e7 + 3.38e7i)T + (-4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-2.80e7 + 4.85e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 2.25e15T^{2} \)
89 \( 1 + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 1.76e8T + 7.83e15T^{2} \)
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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

−12.59764838751501716166067182271, −11.47472227353719282688528811318, −10.45779806734058548792081893487, −8.942597157045577472291852453846, −7.86080792054859724294031077510, −6.33751987369936031392655247574, −5.84939535434579025785834555991, −3.78118840408313968651723801188, −2.11447675848782306995287964914, −0.62229422044988428239712270093, 0.865921114721106077402620347114, 3.23402831653309463547424077618, 4.24309634516056017408683578655, 5.80662300019544935449443762406, 6.74712729227002429321903363654, 8.596079015354438465505077361612, 9.582823229443943774310899158576, 10.70860190258255038628822835500, 11.40306799875168054614033512312, 12.89556064271638119606743822043

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