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Properties

Label 2-84-21.20-c11-0-15
Degree $2$
Conductor $84$
Sign $-0.0840 + 0.996i$
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  + (−300. − 294. i)3-s − 5.02e3·5-s + (−2.83e4 − 3.42e4i)7-s + (3.45e3 + 1.77e5i)9-s + 3.76e5i·11-s + 1.35e5i·13-s + (1.50e6 + 1.48e6i)15-s + 7.23e6·17-s + 5.82e6i·19-s + (−1.57e6 + 1.86e7i)21-s − 1.30e6i·23-s − 2.35e7·25-s + (5.11e7 − 5.42e7i)27-s + 8.52e7i·29-s + 8.26e7i·31-s + ⋯
L(s)  = 1  + (−0.713 − 0.700i)3-s − 0.718·5-s + (−0.637 − 0.770i)7-s + (0.0195 + 0.999i)9-s + 0.704i·11-s + 0.101i·13-s + (0.513 + 0.503i)15-s + 1.23·17-s + 0.539i·19-s + (−0.0840 + 0.996i)21-s − 0.0421i·23-s − 0.483·25-s + (0.686 − 0.727i)27-s + 0.772i·29-s + 0.518i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0840 + 0.996i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.0840 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.0840 + 0.996i$
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: \((2,\ 84,\ (\ :11/2),\ -0.0840 + 0.996i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.544546 - 0.592428i\)
\(L(\frac12)\) \(\approx\) \(0.544546 - 0.592428i\)
\(L(\frac{13}{2})\) 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 \( 1 + (300. + 294. i)T \)
7 \( 1 + (2.83e4 + 3.42e4i)T \)
good5 \( 1 + 5.02e3T + 4.88e7T^{2} \)
11 \( 1 - 3.76e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.35e5iT - 1.79e12T^{2} \)
17 \( 1 - 7.23e6T + 3.42e13T^{2} \)
19 \( 1 - 5.82e6iT - 1.16e14T^{2} \)
23 \( 1 + 1.30e6iT - 9.52e14T^{2} \)
29 \( 1 - 8.52e7iT - 1.22e16T^{2} \)
31 \( 1 - 8.26e7iT - 2.54e16T^{2} \)
37 \( 1 - 5.72e8T + 1.77e17T^{2} \)
41 \( 1 - 2.33e8T + 5.50e17T^{2} \)
43 \( 1 + 1.16e9T + 9.29e17T^{2} \)
47 \( 1 + 2.56e8T + 2.47e18T^{2} \)
53 \( 1 + 1.13e9iT - 9.26e18T^{2} \)
59 \( 1 + 5.04e9T + 3.01e19T^{2} \)
61 \( 1 + 4.16e9iT - 4.35e19T^{2} \)
67 \( 1 + 5.89e9T + 1.22e20T^{2} \)
71 \( 1 + 1.31e10iT - 2.31e20T^{2} \)
73 \( 1 + 3.07e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.61e10T + 7.47e20T^{2} \)
83 \( 1 - 4.28e10T + 1.28e21T^{2} \)
89 \( 1 - 7.80e10T + 2.77e21T^{2} \)
97 \( 1 + 4.16e10iT - 7.15e21T^{2} \)
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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

−11.93382670281090800467123498807, −10.75022499788692213087685218621, −9.771253020968065778663126980815, −7.946585266995551283210751978395, −7.24656866329097856629524585418, −6.12797283841188427213248801717, −4.69861222423831328817238127332, −3.36439306164939576174382616125, −1.54532313490983219676460966765, −0.34916820286410847457290533523, 0.72327872274540459237006313488, 2.94819556302743112503799428268, 4.01099623224256994378862264369, 5.42648717863185345432067736297, 6.31253466647248424484918114990, 7.889673793445394421908028834377, 9.182835153056848703837454595896, 10.12803986721732487063700310618, 11.42246000371559613315856178621, 11.97845384773734306421563841493

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