Properties

Degree 1
Conductor $ 2^{2} \cdot 5^{2} $
Sign $0.0627 - 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.309 + 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (0.809 − 0.587i)23-s + (0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.809 − 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯
L(s,χ)  = 1  + (−0.309 + 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (0.809 − 0.587i)23-s + (0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.809 − 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0627 - 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0627 - 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.0627 - 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{100} (31, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 100,\ (1:\ ),\ 0.0627 - 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4608449620 - 0.4327622248i$
$L(\frac12,\chi)$  $\approx$  $0.4608449620 - 0.4327622248i$
$L(\chi,1)$  $\approx$  0.7371752092 + 0.04637913299i
$L(1,\chi)$  $\approx$  0.7371752092 + 0.04637913299i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−29.563502566082643705808681646508, −29.283621861108947538963220336918, −28.05261319418904798998908232915, −26.870030724653214897204591634118, −25.3636196453880871767195300260, −25.01592319325559569953762120866, −23.55595774476149595621941747832, −22.8173372724719618114590859359, −21.8442014634616648246486865595, −20.145323022743450685270848639687, −19.32141236736612271782480742136, −18.42184684644512107039973227453, −17.133198212650653751092867350160, −16.37978554724940669560305561433, −14.7185416177557305374089236413, −13.634274746232372565858670022206, −12.43666431849570842358704151760, −11.78635466790946841521428118965, −10.11484623580118607695810894943, −8.91173398224218631146361475032, −7.236485812334507357772093424121, −6.59628993716829020219631557170, −5.08834724351264668510780704642, −3.18456918364470671829442409841, −1.560246336727401102574530402334, 0.29022636437210361807260763837, 2.947463480370356448961343047482, 4.126720332424636467799032055066, 5.60256557437483113883434165820, 6.72057165798615857117706644475, 8.61989299945966259778032793596, 9.67278039225319498766397784975, 10.64388593886479702599395688675, 11.901419066263659124349666009514, 13.12932630035721949013112148602, 14.64983897951443198065927235051, 15.52061725834967597949192436065, 16.71975689019295802904888902178, 17.33313305804766079163347434630, 19.133362088459581217234344806444, 19.911056445207523374333679212651, 21.26112397752685396406947392780, 22.19172881622086970056046564509, 22.83439544703337502566509660660, 24.22175256779656038387769883602, 25.5203691062573707028527624533, 26.45035001564266570915879399980, 27.3660442653951492860851023994, 28.3652119654551698465173174414, 29.2812328568194312208501732734

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