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Properties

Label 1-20e2-400.131-r1-0-0
Degree $1$
Conductor $400$
Sign $0.695 - 0.718i$
Analytic cond. $42.9859$
Root an. cond. $42.9859$
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  + (−0.951 − 0.309i)3-s + 7-s + (0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + (0.587 − 0.809i)13-s + (0.309 + 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.951 − 0.309i)21-s + (−0.809 + 0.587i)23-s + (−0.587 − 0.809i)27-s + (0.951 + 0.309i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.587 + 0.809i)37-s + (−0.809 + 0.587i)39-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s + 7-s + (0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + (0.587 − 0.809i)13-s + (0.309 + 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.951 − 0.309i)21-s + (−0.809 + 0.587i)23-s + (−0.587 − 0.809i)27-s + (0.951 + 0.309i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.587 + 0.809i)37-s + (−0.809 + 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(42.9859\)
Root analytic conductor: \(42.9859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (1:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.484544782 - 0.6286235064i\)
\(L(\frac12)\) \(\approx\) \(1.484544782 - 0.6286235064i\)
\(L(1)\) \(\approx\) \(0.9520150942 - 0.1629317142i\)
\(L(1)\) \(\approx\) \(0.9520150942 - 0.1629317142i\)

Euler product

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

−24.11617034665927044526844680048, −23.31721805871995743944939799266, −22.70787472060658396507561479106, −21.5972986045441987877420794992, −20.93130009314558226023112758489, −20.24052089390008988042615366962, −18.65598899101220283411442778534, −18.06067902716687635834091988421, −17.42684167001369677264836506344, −16.20591446326957832106775439909, −15.782924933730435887181985561033, −14.52320667047705841542037167441, −13.73507564219942745311210767610, −12.28248208584769135997128307774, −11.82009949691596591911231311895, −10.8027263804954591390811456233, −10.03735221833852699132957838478, −8.93529996580419240936814966115, −7.65679652960669967709967863719, −6.80967942947553683199909819395, −5.52187103271616619308938836885, −4.84258386889004069532037555110, −3.86844595436857760600507065405, −2.13131556188233508921749133985, −0.91012094652025977763336119571, 0.689140799389995840357667137240, 1.68946688704260681471446908119, 3.28528879761373380147102381705, 4.68185616196162738299138853769, 5.56127885691560211076644267658, 6.28961735761266750959851549990, 7.80608904525774473804939405119, 8.14606522202646712487737181707, 9.80352405214781604228629050787, 10.86801631896856137724085253192, 11.31091277760664313391539746337, 12.38374215254524875554639432922, 13.29891948749586229643108780062, 14.169676534486515058807310247499, 15.4614253573833669150439524076, 16.13942002331778910229362585392, 17.22652244525161706521529011663, 17.92797193439417924430066526773, 18.52132879294550991084842460191, 19.617000649774907997176916580834, 20.779151581230623931233640365727, 21.53828483677467511648760618979, 22.31475398891781639057707752569, 23.36432755721885177856863744926, 24.0007116161260246389567416315

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