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Properties

Label 2-84-28.27-c1-0-3
Degree $2$
Conductor $84$
Sign $0.997 + 0.0636i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 0.599i)2-s + 3-s + (1.28 − 1.53i)4-s − 3.33i·5-s + (−1.28 + 0.599i)6-s + (1.56 + 2.13i)7-s + (−0.719 + 2.73i)8-s + 9-s + (2 + 4.27i)10-s + 0.936i·11-s + (1.28 − 1.53i)12-s − 1.87i·13-s + (−3.28 − 1.79i)14-s − 3.33i·15-s + (−0.719 − 3.93i)16-s + 5.20i·17-s + ⋯
L(s)  = 1  + (−0.905 + 0.424i)2-s + 0.577·3-s + (0.640 − 0.768i)4-s − 1.49i·5-s + (−0.522 + 0.244i)6-s + (0.590 + 0.807i)7-s + (−0.254 + 0.967i)8-s + 0.333·9-s + (0.632 + 1.35i)10-s + 0.282i·11-s + (0.369 − 0.443i)12-s − 0.519i·13-s + (−0.876 − 0.480i)14-s − 0.861i·15-s + (−0.179 − 0.983i)16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.997 + 0.0636i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.997 + 0.0636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802807 - 0.0255711i\)
\(L(\frac12)\) \(\approx\) \(0.802807 - 0.0255711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.599i)T \)
3 \( 1 - T \)
7 \( 1 + (-1.56 - 2.13i)T \)
good5 \( 1 + 3.33iT - 5T^{2} \)
11 \( 1 - 0.936iT - 11T^{2} \)
13 \( 1 + 1.87iT - 13T^{2} \)
17 \( 1 - 5.20iT - 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 + 0.936iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 - 9.06iT - 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 4.79iT - 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 3.86iT - 71T^{2} \)
73 \( 1 + 6.67iT - 73T^{2} \)
79 \( 1 + 2.39iT - 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 1.46iT - 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−14.76819994238536580665720511066, −13.07056018224511956699937647030, −12.22863885781769276542438268423, −10.75320245503232241960206349880, −9.417286413170718152697370458412, −8.510815628725766047549570632269, −8.036450704565657382217270384451, −6.08336522055488476693531156665, −4.73027934549930739366738681298, −1.85079309095007750368930363162, 2.38992235256116385514090455262, 3.85746371001894976005028043319, 6.74425116675161729432485668747, 7.45663452075054569364412035995, 8.718749724391334480388567760508, 10.06560209788034286652920546956, 10.84036931716501584657900716808, 11.67699098899141575774887647947, 13.36373791091753852187686087932, 14.34846655778372350742695271801

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