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Properties

Label 2-84-12.11-c5-0-23
Degree $2$
Conductor $84$
Sign $0.979 - 0.200i$
Analytic cond. $13.4722$
Root an. cond. $3.67045$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.66 − 3.20i)2-s + (15.3 + 2.55i)3-s + (11.4 + 29.8i)4-s − 8.37i·5-s + (−63.5 − 61.1i)6-s + 49i·7-s + (42.0 − 176. i)8-s + (229. + 78.7i)9-s + (−26.8 + 39.0i)10-s + 453.·11-s + (100. + 488. i)12-s − 689.·13-s + (156. − 228. i)14-s + (21.4 − 128. i)15-s + (−759. + 686. i)16-s + 2.11e3i·17-s + ⋯
L(s)  = 1  + (−0.824 − 0.566i)2-s + (0.986 + 0.164i)3-s + (0.359 + 0.933i)4-s − 0.149i·5-s + (−0.720 − 0.693i)6-s + 0.377i·7-s + (0.232 − 0.972i)8-s + (0.946 + 0.323i)9-s + (−0.0848 + 0.123i)10-s + 1.12·11-s + (0.200 + 0.979i)12-s − 1.13·13-s + (0.213 − 0.311i)14-s + (0.0245 − 0.147i)15-s + (−0.742 + 0.670i)16-s + 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.979 - 0.200i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ 0.979 - 0.200i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.76437 + 0.179107i\)
\(L(\frac12)\) \(\approx\) \(1.76437 + 0.179107i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.66 + 3.20i)T \)
3 \( 1 + (-15.3 - 2.55i)T \)
7 \( 1 - 49iT \)
good5 \( 1 + 8.37iT - 3.12e3T^{2} \)
11 \( 1 - 453.T + 1.61e5T^{2} \)
13 \( 1 + 689.T + 3.71e5T^{2} \)
17 \( 1 - 2.11e3iT - 1.41e6T^{2} \)
19 \( 1 - 129. iT - 2.47e6T^{2} \)
23 \( 1 - 4.27e3T + 6.43e6T^{2} \)
29 \( 1 + 4.93e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.62e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.59e3T + 6.93e7T^{2} \)
41 \( 1 - 675. iT - 1.15e8T^{2} \)
43 \( 1 - 1.83e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.04e4T + 2.29e8T^{2} \)
53 \( 1 + 2.33e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.36e4T + 7.14e8T^{2} \)
61 \( 1 + 2.63e4T + 8.44e8T^{2} \)
67 \( 1 + 1.64e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.11e4T + 1.80e9T^{2} \)
73 \( 1 - 1.83e4T + 2.07e9T^{2} \)
79 \( 1 - 8.46e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.43e4T + 3.93e9T^{2} \)
89 \( 1 + 1.17e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.13e5T + 8.58e9T^{2} \)
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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

−13.03572055254735068427078128026, −12.31640178588476648075762148954, −10.95105856910606139309060297950, −9.750022604619014689021021493792, −8.977411401599375123713806211162, −8.029024362215871895070243771087, −6.74519603155264523005466903081, −4.30995640228323175718390032149, −2.89875800665064296402284405232, −1.46491708277735980410321723261, 1.00694290792320026213412330618, 2.77379554800675668262170541420, 4.83035712695565756541298253118, 6.93409937736225467967511657302, 7.34161834299221166708665939713, 8.961065668812671263022791802185, 9.444226465559637482023956185259, 10.76341789306733743254625194084, 12.14796457179370208429386354071, 13.70862067856135357742011737220

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