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Properties

Label 2-84-84.11-c1-0-5
Degree $2$
Conductor $84$
Sign $0.470 - 0.882i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.430 + 1.34i)2-s + (1.66 − 0.475i)3-s + (−1.62 + 1.16i)4-s + (−2.36 + 1.36i)5-s + (1.35 + 2.03i)6-s + (1.89 − 1.84i)7-s + (−2.26 − 1.69i)8-s + (2.54 − 1.58i)9-s + (−2.85 − 2.59i)10-s + (1.02 − 1.76i)11-s + (−2.16 + 2.70i)12-s − 4.44·13-s + (3.30 + 1.76i)14-s + (−3.28 + 3.39i)15-s + (1.30 − 3.78i)16-s + (0.571 + 0.330i)17-s + ⋯
L(s)  = 1  + (0.304 + 0.952i)2-s + (0.961 − 0.274i)3-s + (−0.814 + 0.580i)4-s + (−1.05 + 0.609i)5-s + (0.554 + 0.832i)6-s + (0.717 − 0.696i)7-s + (−0.800 − 0.598i)8-s + (0.849 − 0.528i)9-s + (−0.902 − 0.820i)10-s + (0.307 − 0.532i)11-s + (−0.623 + 0.781i)12-s − 1.23·13-s + (0.882 + 0.470i)14-s + (−0.848 + 0.876i)15-s + (0.326 − 0.945i)16-s + (0.138 + 0.0800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.470 - 0.882i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.470 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03059 + 0.618729i\)
\(L(\frac12)\) \(\approx\) \(1.03059 + 0.618729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.430 - 1.34i)T \)
3 \( 1 + (-1.66 + 0.475i)T \)
7 \( 1 + (-1.89 + 1.84i)T \)
good5 \( 1 + (2.36 - 1.36i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.02 + 1.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + (-0.571 - 0.330i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.54 - 1.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.521 - 0.902i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.06iT - 29T^{2} \)
31 \( 1 + (2.68 + 1.55i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.76 - 8.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.85iT - 41T^{2} \)
43 \( 1 - 0.530iT - 43T^{2} \)
47 \( 1 + (0.521 + 0.902i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.16 + 5.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.74 + 3.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0992 - 0.171i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.12 - 2.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 + (-1.51 + 2.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.45 + 5.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + (-0.468 + 0.270i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.91T + 97T^{2} \)
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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

−14.76691263334051403635247361887, −13.78216074637068769721169592851, −12.62193131198889875960858854665, −11.40985930725962930870922521581, −9.774526147657221815494957126512, −8.278514228804697814637100588917, −7.66675023732866860190111930341, −6.73438879844264322234114852194, −4.55412580941567239338883939063, −3.38181013523111333136938721357, 2.31483020948501690108062067824, 4.08886205109555417192187641363, 4.98042309321659557654782873770, 7.64682221618469336801412696201, 8.710245075501310838134390962834, 9.564245622911418460748883745319, 10.97119200727352152313391028288, 12.17121389381408032280928605663, 12.68791214896377896694840597034, 14.22541413280340200094827017155

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