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Properties

Label 2-84-21.20-c11-0-26
Degree $2$
Conductor $84$
Sign $-0.355 + 0.934i$
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  + (−419. + 35.3i)3-s + 9.42e3·5-s + (1.92e4 − 4.00e4i)7-s + (1.74e5 − 2.96e4i)9-s − 9.05e5i·11-s − 9.42e5i·13-s + (−3.95e6 + 3.33e5i)15-s + 5.13e6·17-s − 6.72e6i·19-s + (−6.65e6 + 1.74e7i)21-s − 1.46e7i·23-s + 4.00e7·25-s + (−7.21e7 + 1.86e7i)27-s + 4.78e7i·29-s + 2.52e8i·31-s + ⋯
L(s)  = 1  + (−0.996 + 0.0840i)3-s + 1.34·5-s + (0.432 − 0.901i)7-s + (0.985 − 0.167i)9-s − 1.69i·11-s − 0.704i·13-s + (−1.34 + 0.113i)15-s + 0.877·17-s − 0.623i·19-s + (−0.355 + 0.934i)21-s − 0.475i·23-s + 0.820·25-s + (−0.968 + 0.249i)27-s + 0.432i·29-s + 1.58i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.355 + 0.934i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.355 + 0.934i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.08495 - 1.57330i\)
\(L(\frac12)\) \(\approx\) \(1.08495 - 1.57330i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (419. - 35.3i)T \)
7 \( 1 + (-1.92e4 + 4.00e4i)T \)
good5 \( 1 - 9.42e3T + 4.88e7T^{2} \)
11 \( 1 + 9.05e5iT - 2.85e11T^{2} \)
13 \( 1 + 9.42e5iT - 1.79e12T^{2} \)
17 \( 1 - 5.13e6T + 3.42e13T^{2} \)
19 \( 1 + 6.72e6iT - 1.16e14T^{2} \)
23 \( 1 + 1.46e7iT - 9.52e14T^{2} \)
29 \( 1 - 4.78e7iT - 1.22e16T^{2} \)
31 \( 1 - 2.52e8iT - 2.54e16T^{2} \)
37 \( 1 - 6.12e8T + 1.77e17T^{2} \)
41 \( 1 + 6.04e8T + 5.50e17T^{2} \)
43 \( 1 + 2.28e8T + 9.29e17T^{2} \)
47 \( 1 + 2.41e9T + 2.47e18T^{2} \)
53 \( 1 + 5.67e9iT - 9.26e18T^{2} \)
59 \( 1 - 1.15e9T + 3.01e19T^{2} \)
61 \( 1 - 9.53e9iT - 4.35e19T^{2} \)
67 \( 1 - 3.19e9T + 1.22e20T^{2} \)
71 \( 1 + 2.41e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.89e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.36e10T + 7.47e20T^{2} \)
83 \( 1 - 1.59e10T + 1.28e21T^{2} \)
89 \( 1 + 4.14e10T + 2.77e21T^{2} \)
97 \( 1 - 4.99e10iT - 7.15e21T^{2} \)
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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

−11.43592942472864380898944944928, −10.59552501094330760301905071460, −9.867839910961181647995318044381, −8.344565754095757746064249182017, −6.80848732883711600169851865123, −5.80790314090386813161972605357, −4.97994924885636827107071057910, −3.26298869992274430239242398688, −1.39524186356163063765323034741, −0.55644299421713845031268575600, 1.48641099774057398327661226794, 2.16401728187667140079589604899, 4.51316953946211752987977342734, 5.53948303160785992113566136706, 6.34431886751030572734113436399, 7.68586479904388114779486226835, 9.583543545948455099965382092575, 9.901761163346935076493646463813, 11.43297105480629203895208965039, 12.29082245280599074849845937171

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