↑

Properties

Label 1-84-84.59-r1-0-0
Degree $1$
Conductor $84$
Sign $-0.991 - 0.126i$
Analytic cond. $9.02705$
Root an. cond. $9.02705$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)11-s − 13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 41-s − 43-s + (0.5 − 0.866i)47-s + (0.5 + 0.866i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)11-s − 13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 41-s − 43-s + (0.5 − 0.866i)47-s + (0.5 + 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(9.02705\)
Root analytic conductor: \(9.02705\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 84,\ (1:\ ),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006754813381 + 0.1064417541i\)
\(L(\frac12)\) \(\approx\) \(0.006754813381 + 0.1064417541i\)
\(L(1)\) \(\approx\) \(0.6800521676 + 0.08666147500i\)
\(L(1)\) \(\approx\) \(0.6800521676 + 0.08666147500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.09296175832345639199155775443, −28.65921755634291119545043499734, −28.099491438132979578758630321215, −26.86331676015131759196298587798, −25.78823321894642643506319547038, −24.46922636919233841358942839252, −23.7779853120497779139569344696, −22.54765733549928465050606102517, −21.305418146443683504065837110708, −20.14869875112966893968664001674, −19.419762110180769379467468843422, −17.852319489178950777930957223455, −16.87114664580238510006331928516, −15.67150163533345167330992235800, −14.68632243733093243005915353251, −12.96616556643733607538851601057, −12.33242283594329911685870217437, −10.845546833591945450101878187493, −9.45249196327773318171175531821, −8.24501634381663701308683613511, −7.02048814094043550062814007489, −5.20410232001639393310358722381, −4.150675981361680424033891692212, −2.125351501808838728582535582363, −0.04679959096731799967284018353, 2.46845435737195258379327043307, 3.837389875079354389370214599208, 5.566062373968980361548994344431, 7.042766592911424033333627921150, 8.09189499551442968009325324501, 9.73173819572569088801931885433, 10.95208971904920178626871227717, 11.93862849492322997468076996313, 13.468581525233280518369833769850, 14.61502516102634747192292200952, 15.636932014297276923314171481569, 16.84995099555567988036650204426, 18.28238996385279043486707097944, 19.06418132110776230625937436484, 20.2128353107232475339331239671, 21.62518396146473215930300063324, 22.49470919995170236391305571967, 23.60447507797590863350105021056, 24.64021827284966783671935212404, 26.00743259798001839000742870384, 26.8644839995237322297507010017, 27.69773568100531439319707441164, 29.34777381821147052864462108326, 29.80766252183609808358544132990, 31.368674511005381227840381420673

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