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Properties

Label 2-84-84.23-c1-0-2
Degree $2$
Conductor $84$
Sign $0.0231 - 0.999i$
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  + (0.162 + 1.40i)2-s + (1.27 + 1.16i)3-s + (−1.94 + 0.455i)4-s + (−0.432 − 0.249i)5-s + (−1.43 + 1.98i)6-s + (0.261 − 2.63i)7-s + (−0.956 − 2.66i)8-s + (0.267 + 2.98i)9-s + (0.280 − 0.648i)10-s + (−0.695 − 1.20i)11-s + (−3.02 − 1.69i)12-s + 2.75·13-s + (3.74 − 0.0590i)14-s + (−0.260 − 0.824i)15-s + (3.58 − 1.77i)16-s + (−5.04 + 2.91i)17-s + ⋯
L(s)  = 1  + (0.114 + 0.993i)2-s + (0.737 + 0.674i)3-s + (−0.973 + 0.227i)4-s + (−0.193 − 0.111i)5-s + (−0.585 + 0.810i)6-s + (0.0989 − 0.995i)7-s + (−0.338 − 0.941i)8-s + (0.0890 + 0.996i)9-s + (0.0887 − 0.204i)10-s + (−0.209 − 0.363i)11-s + (−0.872 − 0.488i)12-s + 0.762·13-s + (0.999 − 0.0157i)14-s + (−0.0673 − 0.212i)15-s + (0.896 − 0.443i)16-s + (−1.22 + 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.0231 - 0.999i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.0231 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788389 + 0.770362i\)
\(L(\frac12)\) \(\approx\) \(0.788389 + 0.770362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.162 - 1.40i)T \)
3 \( 1 + (-1.27 - 1.16i)T \)
7 \( 1 + (-0.261 + 2.63i)T \)
good5 \( 1 + (0.432 + 0.249i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.695 + 1.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (5.04 - 2.91i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.14 - 1.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.53 + 4.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.32iT - 29T^{2} \)
31 \( 1 + (5.98 - 3.45i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.67 - 6.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.74iT - 41T^{2} \)
43 \( 1 - 3.52iT - 43T^{2} \)
47 \( 1 + (2.53 - 4.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.54 + 2.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.57 + 2.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.45 + 4.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.30 + 5.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 + (-4.98 - 8.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.84 + 1.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + (-2.12 - 1.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.526T + 97T^{2} \)
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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

−14.58961707960004590152316513700, −13.72146423778434865464689720553, −13.01101503812042437864056288344, −11.01765081921313944437884134387, −9.935919392164242467675630520271, −8.637886833757368948621860028137, −7.913858572424119656778549784247, −6.47239813196004180447864418066, −4.71458667930976789739063212834, −3.68218275236698978385342598332, 2.07785762547226217750526735487, 3.51849637345732126953952247071, 5.43626071725075186402313676974, 7.27736517081650721122986051614, 8.793718716008160525832426600010, 9.299615706428294978467253269430, 11.04312337697892120283798434806, 11.92169030951379139055096789223, 12.97063553784471282758519417137, 13.67544844188022750616525930567

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