Properties

Label 2-84-28.27-c1-0-5
Degree $2$
Conductor $84$
Sign $-0.242 + 0.970i$
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 + (0.719 + 3.67i)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 + (0.192 + 0.981i)14-s + 0.861i·15-s + (−0.179 + 0.983i)16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.242 + 0.970i$
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: \((2,\ 84,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315661 - 0.404080i\)
\(L(\frac12)\) \(\approx\) \(0.315661 - 0.404080i\)
\(L(\frac{3}{2})\) 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

−13.45024545278687333097783788502, −12.70023491209255873490481778914, −11.80808477627811878131244733606, −10.54090715318362925293792215804, −9.615539218424547144561418829750, −8.485088658869101834459097857355, −7.30083954693586896562771172570, −5.65037187613062237570453708006, −3.85272822630576510105703056642, −0.985351524338355910375931631778, 2.74612645231323554418663994176, 5.53586871994434092366007764959, 6.69264289318017612243495303152, 7.43224542851742676728711725785, 9.284772975050576703196082624885, 10.05605089236899779108555143567, 11.26048853340563633005253091148, 11.93192766094129542846043672105, 13.84321846689656638694858760498, 14.86608791623833703678971231846

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