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Properties

Label 1-10e2-100.71-r1-0-0
Degree $1$
Conductor $100$
Sign $0.0627 + 0.998i$
Analytic cond. $10.7464$
Root an. cond. $10.7464$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.0627 + 0.998i$
Analytic conductor: \(10.7464\)
Root analytic conductor: \(10.7464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 100,\ (1:\ ),\ 0.0627 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4608449620 + 0.4327622248i\)
\(L(\frac12)\) \(\approx\) \(0.4608449620 + 0.4327622248i\)
\(L(1)\) \(\approx\) \(0.7371752092 - 0.04637913299i\)
\(L(1)\) \(\approx\) \(0.7371752092 - 0.04637913299i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 - T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

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

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