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Properties

Label 1-84-84.83-r1-0-0
Degree $1$
Conductor $84$
Sign $1$
Analytic cond. $9.02705$
Root an. cond. $9.02705$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 11-s − 13-s + 17-s + 19-s + 23-s + 25-s − 29-s + 31-s + 37-s + 41-s − 43-s − 47-s − 53-s + 55-s − 59-s − 61-s − 65-s − 67-s + 71-s − 73-s − 79-s − 83-s + 85-s + 89-s + 95-s − 97-s + ⋯
L(s)  = 1  + 5-s + 11-s − 13-s + 17-s + 19-s + 23-s + 25-s − 29-s + 31-s + 37-s + 41-s − 43-s − 47-s − 53-s + 55-s − 59-s − 61-s − 65-s − 67-s + 71-s − 73-s − 79-s − 83-s + 85-s + 89-s + 95-s − 97-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(9.02705\)
Root analytic conductor: \(9.02705\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{84} (83, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 84,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.071615032\)
\(L(\frac12)\) \(\approx\) \(2.071615032\)
\(L(1)\) \(\approx\) \(1.371103441\)
\(L(1)\) \(\approx\) \(1.371103441\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.318875564549180853368386098162, −29.55527792667662469047433594106, −28.56451441372783857705933147920, −27.36540364370798490866626261595, −26.28805339384335928892718648440, −25.06668453964710064030385002601, −24.49671752237924488889623444476, −22.86919645606075561776957014307, −21.954333497290583379408436281032, −20.96637993057771093227446515383, −19.755194055226652736800219425465, −18.54565512368966805894765348170, −17.31794301098718752362651170736, −16.60621427578771813595140102910, −14.8605333649607761986756344376, −14.03394697745788555108836305297, −12.758892954514545739322108965678, −11.531622011391680830969979734547, −9.96733005366762894042922552044, −9.2173425502966721991056101026, −7.47709285520385098954142705507, −6.14563887156622542686060591156, −4.885738652161477809477623520971, −3.01290319592163793308748231112, −1.34427552111842394956762630018, 1.34427552111842394956762630018, 3.01290319592163793308748231112, 4.885738652161477809477623520971, 6.14563887156622542686060591156, 7.47709285520385098954142705507, 9.2173425502966721991056101026, 9.96733005366762894042922552044, 11.531622011391680830969979734547, 12.758892954514545739322108965678, 14.03394697745788555108836305297, 14.8605333649607761986756344376, 16.60621427578771813595140102910, 17.31794301098718752362651170736, 18.54565512368966805894765348170, 19.755194055226652736800219425465, 20.96637993057771093227446515383, 21.954333497290583379408436281032, 22.86919645606075561776957014307, 24.49671752237924488889623444476, 25.06668453964710064030385002601, 26.28805339384335928892718648440, 27.36540364370798490866626261595, 28.56451441372783857705933147920, 29.55527792667662469047433594106, 30.318875564549180853368386098162

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