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Properties

Label 1-10e2-100.79-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: $\chi_{100} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 100,\ (1:\ ),\ 0.0627 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.496672563 + 1.405469088i\)
\(L(\frac12)\) \(\approx\) \(1.496672563 + 1.405469088i\)
\(L(1)\) \(\approx\) \(1.232226991 + 0.5520356740i\)
\(L(1)\) \(\approx\) \(1.232226991 + 0.5520356740i\)

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

−29.70179068212558991858256055640, −28.48977587554544865579579206229, −27.39842368966544391737396701680, −26.23104516453882658682039714493, −25.13572623122384216771573988670, −24.223839376274525634776693920155, −23.55643445746072216690068575453, −22.13694979919809783678840825521, −20.89437516621593388075376027965, −19.89952965303485321505575495335, −18.79221237467406943996833947510, −17.91019518041715484898818987909, −16.89781334972209884359811100360, −15.330394021210494169091770252780, −14.037555493140191143877127052708, −13.501527673600670961517934864268, −11.8064874966569914626161514129, −11.273102177754058236377513803833, −9.20368625363715182261765811684, −8.272876029680603935534272223791, −7.05566098311095027172535027458, −5.85821055704741978902469288453, −4.09931756726872928661928901431, −2.336746970799114484203155694647, −0.96571343473865083833130206369, 1.767833882594145382899210899180, 3.64163179465153110049981934980, 4.66904180412287706543862698610, 6.07939431900958340068805099698, 7.98338912289683342198169437500, 8.84642071513005989020576287774, 10.265951497685155114144105253642, 11.11215152589730242123969605364, 12.50788632192964946437581504273, 14.205143268258181715154736540379, 14.790687531104711998382877738095, 15.96553654983800279982456743571, 17.12351617555178829398989312133, 18.13327507520039596381584909196, 19.74231505860460231749133842331, 20.54295465527892004308478448557, 21.479487856027351473103541491920, 22.47130613006915796524632183160, 23.62546922179372212359129198105, 24.99639556023058020011655438709, 25.79026336257016687858907915230, 27.07949949272555408131758898879, 27.68486288873703752784423555483, 28.5807105886909724981713833062, 30.30862264419532640293262280721

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