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Properties

Label 2-84-12.11-c5-0-21
Degree $2$
Conductor $84$
Sign $-0.0514 - 0.998i$
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  + (−3.60 + 4.36i)2-s + (−2.15 + 15.4i)3-s + (−6.05 − 31.4i)4-s − 60.5i·5-s + (−59.5 − 65.0i)6-s + 49i·7-s + (158. + 86.7i)8-s + (−233. − 66.5i)9-s + (264. + 218. i)10-s + 697.·11-s + (498. − 25.6i)12-s + 189.·13-s + (−213. − 176. i)14-s + (934. + 130. i)15-s + (−950. + 380. i)16-s − 419. i·17-s + ⋯
L(s)  = 1  + (−0.636 + 0.771i)2-s + (−0.138 + 0.990i)3-s + (−0.189 − 0.981i)4-s − 1.08i·5-s + (−0.675 − 0.737i)6-s + 0.377i·7-s + (0.877 + 0.479i)8-s + (−0.961 − 0.274i)9-s + (0.835 + 0.689i)10-s + 1.73·11-s + (0.998 − 0.0514i)12-s + 0.311·13-s + (−0.291 − 0.240i)14-s + (1.07 + 0.149i)15-s + (−0.928 + 0.371i)16-s − 0.351i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0514 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0514 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.0514 - 0.998i$
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: \((2,\ 84,\ (\ :5/2),\ -0.0514 - 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.845969 + 0.890714i\)
\(L(\frac12)\) \(\approx\) \(0.845969 + 0.890714i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.60 - 4.36i)T \)
3 \( 1 + (2.15 - 15.4i)T \)
7 \( 1 - 49iT \)
good5 \( 1 + 60.5iT - 3.12e3T^{2} \)
11 \( 1 - 697.T + 1.61e5T^{2} \)
13 \( 1 - 189.T + 3.71e5T^{2} \)
17 \( 1 + 419. iT - 1.41e6T^{2} \)
19 \( 1 - 1.89e3iT - 2.47e6T^{2} \)
23 \( 1 + 589.T + 6.43e6T^{2} \)
29 \( 1 - 1.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.67e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.14e4T + 6.93e7T^{2} \)
41 \( 1 + 717. iT - 1.15e8T^{2} \)
43 \( 1 - 7.80e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.70e3T + 2.29e8T^{2} \)
53 \( 1 - 2.54e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.74e4T + 7.14e8T^{2} \)
61 \( 1 - 2.52e4T + 8.44e8T^{2} \)
67 \( 1 + 5.30e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.37e4T + 1.80e9T^{2} \)
73 \( 1 - 5.88e4T + 2.07e9T^{2} \)
79 \( 1 - 7.38e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.27e4T + 3.93e9T^{2} \)
89 \( 1 + 1.01e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.46e5T + 8.58e9T^{2} \)
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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

−14.08242395185165387756416883500, −12.33782017716751329390074724336, −11.23181537750882690261248062022, −9.816014030739595642020780582258, −9.077746373430728753138199170384, −8.319886394346975062194456434214, −6.41936903324274852523888893362, −5.30075737068808335343073004753, −4.09546841215764554983837162853, −1.14781117750863802795734198282, 0.835045688503849014219927891417, 2.35484528161638255040368045021, 3.83458813936068047064090385623, 6.45904629044014448308216835371, 7.20449891306249539948402336907, 8.515593300732691106529516100557, 9.772161608698479866559282812590, 11.22717696478656087588575022698, 11.52581820210158371791847854440, 12.89956298219521311708702349084

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