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Properties

Label 2-84-28.3-c1-0-7
Degree $2$
Conductor $84$
Sign $-0.112 + 0.993i$
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  + (−0.272 − 1.38i)2-s + (0.5 − 0.866i)3-s + (−1.85 + 0.755i)4-s + (2.12 − 1.22i)5-s + (−1.33 − 0.458i)6-s + (−2.63 − 0.272i)7-s + (1.55 + 2.36i)8-s + (−0.499 − 0.866i)9-s + (−2.27 − 2.61i)10-s + (1.09 + 0.632i)11-s + (−0.272 + 1.98i)12-s + 2.99i·13-s + (0.337 + 3.72i)14-s − 2.45i·15-s + (2.85 − 2.79i)16-s + (1.58 + 0.916i)17-s + ⋯
L(s)  = 1  + (−0.192 − 0.981i)2-s + (0.288 − 0.499i)3-s + (−0.925 + 0.377i)4-s + (0.949 − 0.548i)5-s + (−0.546 − 0.187i)6-s + (−0.994 − 0.102i)7-s + (0.548 + 0.836i)8-s + (−0.166 − 0.288i)9-s + (−0.720 − 0.826i)10-s + (0.330 + 0.190i)11-s + (−0.0785 + 0.571i)12-s + 0.831i·13-s + (0.0903 + 0.995i)14-s − 0.633i·15-s + (0.714 − 0.699i)16-s + (0.385 + 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.112 + 0.993i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ -0.112 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.623795 - 0.698480i\)
\(L(\frac12)\) \(\approx\) \(0.623795 - 0.698480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.272 + 1.38i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.63 + 0.272i)T \)
good5 \( 1 + (-2.12 + 1.22i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.09 - 0.632i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.99iT - 13T^{2} \)
17 \( 1 + (-1.58 - 0.916i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.07 - 3.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.83 + 3.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + (4.71 - 8.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.75 + 6.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.08iT - 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 + (3.67 + 6.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0358 + 0.0620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.68 - 2.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.61 - 5.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.43 - 1.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (-7.01 - 4.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.54 + 0.891i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.33T + 83T^{2} \)
89 \( 1 + (-7.42 + 4.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.10iT - 97T^{2} \)
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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

−13.64219102087776478565736680040, −12.83753833012265015387586461340, −12.13313445455123252625626580900, −10.60530040951209641558283691953, −9.428703454992446198819144429363, −8.918471793565344496158755205524, −7.12929517937274015336582788531, −5.50388457154688838718602866810, −3.57245235624970420062705961273, −1.75982714799256931346701596506, 3.29935332341078946593648610885, 5.32764061068851061658869865527, 6.35145197082176472332724945489, 7.61346004656520709271251716744, 9.329391359995203186891067852672, 9.635931531678461452073151778625, 10.93282878745593225050959354415, 13.02459364533862348102812488053, 13.62668406171228007377086593734, 14.79368577249822585861760454752

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