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Properties

Label 2-84-28.27-c11-0-55
Degree $2$
Conductor $84$
Sign $0.938 - 0.346i$
Analytic cond. $64.5408$
Root an. cond. $8.03373$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (45.1 + 3.52i)2-s − 243·3-s + (2.02e3 + 317. i)4-s − 1.27e3i·5-s + (−1.09e4 − 855. i)6-s + (3.88e4 − 2.16e4i)7-s + (9.01e4 + 2.14e4i)8-s + 5.90e4·9-s + (4.50e3 − 5.76e4i)10-s + 1.27e5i·11-s + (−4.91e5 − 7.72e4i)12-s + 2.43e6i·13-s + (1.82e6 − 8.41e5i)14-s + 3.10e5i·15-s + (3.99e6 + 1.28e6i)16-s − 2.21e6i·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0777i)2-s − 0.577·3-s + (0.987 + 0.155i)4-s − 0.182i·5-s + (−0.575 − 0.0449i)6-s + (0.873 − 0.487i)7-s + (0.972 + 0.231i)8-s + 0.333·9-s + (0.0142 − 0.182i)10-s + 0.239i·11-s + (−0.570 − 0.0895i)12-s + 1.81i·13-s + (0.908 − 0.418i)14-s + 0.105i·15-s + (0.951 + 0.306i)16-s − 0.378i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.938 - 0.346i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ 0.938 - 0.346i)\)

Particular Values

\(L(6)\) \(\approx\) \(4.08573 + 0.729485i\)
\(L(\frac12)\) \(\approx\) \(4.08573 + 0.729485i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-45.1 - 3.52i)T \)
3 \( 1 + 243T \)
7 \( 1 + (-3.88e4 + 2.16e4i)T \)
good5 \( 1 + 1.27e3iT - 4.88e7T^{2} \)
11 \( 1 - 1.27e5iT - 2.85e11T^{2} \)
13 \( 1 - 2.43e6iT - 1.79e12T^{2} \)
17 \( 1 + 2.21e6iT - 3.42e13T^{2} \)
19 \( 1 + 9.19e6T + 1.16e14T^{2} \)
23 \( 1 + 5.31e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.00e8T + 1.22e16T^{2} \)
31 \( 1 + 2.39e7T + 2.54e16T^{2} \)
37 \( 1 - 6.67e7T + 1.77e17T^{2} \)
41 \( 1 - 1.03e9iT - 5.50e17T^{2} \)
43 \( 1 - 1.48e9iT - 9.29e17T^{2} \)
47 \( 1 - 1.46e9T + 2.47e18T^{2} \)
53 \( 1 - 3.45e9T + 9.26e18T^{2} \)
59 \( 1 - 7.20e9T + 3.01e19T^{2} \)
61 \( 1 - 7.70e9iT - 4.35e19T^{2} \)
67 \( 1 - 3.53e8iT - 1.22e20T^{2} \)
71 \( 1 - 1.06e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.55e10iT - 3.13e20T^{2} \)
79 \( 1 + 4.72e10iT - 7.47e20T^{2} \)
83 \( 1 + 6.02e9T + 1.28e21T^{2} \)
89 \( 1 + 2.15e10iT - 2.77e21T^{2} \)
97 \( 1 + 4.41e10iT - 7.15e21T^{2} \)
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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

−12.05412466876411199739200886208, −11.29349951313464097022079168219, −10.34470569203551163819991536408, −8.580231229829897652817619315466, −7.12119846203301952899347549519, −6.32436411313599296185185730158, −4.68094705909378766816284742768, −4.36634722524419290240481198997, −2.35263307601231571856193254133, −1.12086122543078418839840539976, 0.926154747375676696123696307950, 2.33596698857480160062934548473, 3.70751162256537471346275007064, 5.18429497067351150144251389588, 5.75543153684042297672964375321, 7.18962055323817087925122410841, 8.371254757799950010579507347437, 10.35855477093424667614350197583, 11.00687901208777206599419811286, 12.08766587890432725483809326672

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