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Properties

Label 2-84-12.11-c1-0-11
Degree $2$
Conductor $84$
Sign $0.189 + 0.981i$
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  + (0.750 − 1.19i)2-s + (0.448 − 1.67i)3-s + (−0.874 − 1.79i)4-s + 3.56i·5-s + (−1.66 − 1.79i)6-s + i·7-s + (−2.81 − 0.301i)8-s + (−2.59 − 1.50i)9-s + (4.27 + 2.67i)10-s − 0.335·11-s + (−3.40 + 0.655i)12-s + 3.34·13-s + (1.19 + 0.750i)14-s + (5.96 + 1.59i)15-s + (−2.47 + 3.14i)16-s + 0.335i·17-s + ⋯
L(s)  = 1  + (0.530 − 0.847i)2-s + (0.258 − 0.965i)3-s + (−0.437 − 0.899i)4-s + 1.59i·5-s + (−0.681 − 0.731i)6-s + 0.377i·7-s + (−0.994 − 0.106i)8-s + (−0.865 − 0.500i)9-s + (1.35 + 0.845i)10-s − 0.101·11-s + (−0.981 + 0.189i)12-s + 0.928·13-s + (0.320 + 0.200i)14-s + (1.53 + 0.412i)15-s + (−0.617 + 0.786i)16-s + 0.0813i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.189 + 0.981i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.940017 - 0.776134i\)
\(L(\frac12)\) \(\approx\) \(0.940017 - 0.776134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.750 + 1.19i)T \)
3 \( 1 + (-0.448 + 1.67i)T \)
7 \( 1 - iT \)
good5 \( 1 - 3.56iT - 5T^{2} \)
11 \( 1 + 0.335T + 11T^{2} \)
13 \( 1 - 3.34T + 13T^{2} \)
17 \( 1 - 0.335iT - 17T^{2} \)
19 \( 1 + 1.84iT - 19T^{2} \)
23 \( 1 + 4.45T + 23T^{2} \)
29 \( 1 + 5.91iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 - 1.45iT - 41T^{2} \)
43 \( 1 + 7.49iT - 43T^{2} \)
47 \( 1 + 8.91T + 47T^{2} \)
53 \( 1 + 4.79iT - 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 0.353T + 61T^{2} \)
67 \( 1 - 3.19iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 4.69T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 6.89T + 83T^{2} \)
89 \( 1 + 3.87iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−13.93355871930954896110904918693, −13.06993669972082585047983374178, −11.79501487953859952550238008702, −11.11037154355871987431109699800, −9.939109350314260444833224335925, −8.398963881976072422425420213824, −6.81280509332250119822869063759, −5.89945417986949285636535797622, −3.49805639428631001786748240718, −2.28496944310686873363744545772, 3.80992266774595779152159052255, 4.80676815139854648064208049283, 5.91299378196960688818271448792, 7.975717431978972952439262347030, 8.722520243832338984390305706054, 9.754826164854288600859256925489, 11.44737623021923705938010315861, 12.73017392197344045234600496747, 13.56531951055440061903874008996, 14.56746585428248174187061042756

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