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Properties

Label 2-84-4.3-c2-0-4
Degree $2$
Conductor $84$
Sign $0.582 - 0.812i$
Analytic cond. $2.28883$
Root an. cond. $1.51288$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 + 1.77i)2-s − 1.73i·3-s + (−2.33 + 3.25i)4-s + 8.22·5-s + (3.08 − 1.58i)6-s + 2.64i·7-s + (−7.91 − 1.17i)8-s − 2.99·9-s + (7.51 + 14.6i)10-s + 4.58i·11-s + (5.63 + 4.03i)12-s + 10.9·13-s + (−4.70 + 2.41i)14-s − 14.2i·15-s + (−5.13 − 15.1i)16-s − 30.4·17-s + ⋯
L(s)  = 1  + (0.456 + 0.889i)2-s − 0.577i·3-s + (−0.582 + 0.812i)4-s + 1.64·5-s + (0.513 − 0.263i)6-s + 0.377i·7-s + (−0.989 − 0.147i)8-s − 0.333·9-s + (0.751 + 1.46i)10-s + 0.416i·11-s + (0.469 + 0.336i)12-s + 0.844·13-s + (−0.336 + 0.172i)14-s − 0.950i·15-s + (−0.321 − 0.947i)16-s − 1.79·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.582 - 0.812i$
Analytic conductor: \(2.28883\)
Root analytic conductor: \(1.51288\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1),\ 0.582 - 0.812i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.58117 + 0.811991i\)
\(L(\frac12)\) \(\approx\) \(1.58117 + 0.811991i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.913 - 1.77i)T \)
3 \( 1 + 1.73iT \)
7 \( 1 - 2.64iT \)
good5 \( 1 - 8.22T + 25T^{2} \)
11 \( 1 - 4.58iT - 121T^{2} \)
13 \( 1 - 10.9T + 169T^{2} \)
17 \( 1 + 30.4T + 289T^{2} \)
19 \( 1 + 18.9iT - 361T^{2} \)
23 \( 1 + 37.7iT - 529T^{2} \)
29 \( 1 + 22.8T + 841T^{2} \)
31 \( 1 - 17.1iT - 961T^{2} \)
37 \( 1 + 4.34T + 1.36e3T^{2} \)
41 \( 1 + 11.8T + 1.68e3T^{2} \)
43 \( 1 - 22.1iT - 1.84e3T^{2} \)
47 \( 1 - 55.7iT - 2.20e3T^{2} \)
53 \( 1 + 40.2T + 2.80e3T^{2} \)
59 \( 1 - 21.0iT - 3.48e3T^{2} \)
61 \( 1 - 74.5T + 3.72e3T^{2} \)
67 \( 1 + 16.0iT - 4.48e3T^{2} \)
71 \( 1 + 69.1iT - 5.04e3T^{2} \)
73 \( 1 - 36.1T + 5.32e3T^{2} \)
79 \( 1 - 119. iT - 6.24e3T^{2} \)
83 \( 1 - 25.9iT - 6.88e3T^{2} \)
89 \( 1 - 144.T + 7.92e3T^{2} \)
97 \( 1 - 129.T + 9.40e3T^{2} \)
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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

−14.02707217185151161324806456857, −13.26761250358466967045863013780, −12.66449557027781391463240164955, −10.99724995911524542818756517378, −9.335125948004392890141645874290, −8.576552681319538928674331814655, −6.77518087020506574350753942725, −6.17174369896681258628123987206, −4.83709541594587752145677306780, −2.40350246587486641820987436725, 1.90477972066787245772958305666, 3.72457680328923762346146643308, 5.33665332126294461451887004285, 6.27346393494905416088609743843, 8.825531970695266529882553585382, 9.682586983003327812182385984133, 10.58832094694639737314338540436, 11.45680781646567142592295105185, 13.26882092409695034052805589380, 13.49636253545399861043993599542

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