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Properties

Label 1-20e2-400.291-r1-0-0
Degree $1$
Conductor $400$
Sign $-0.780 + 0.625i$
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  + (0.587 + 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)17-s + (−0.587 + 0.809i)19-s + (0.587 + 0.809i)21-s + (0.309 + 0.951i)23-s + (−0.951 + 0.309i)27-s + (−0.587 − 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.951 − 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)17-s + (−0.587 + 0.809i)19-s + (0.587 + 0.809i)21-s + (0.309 + 0.951i)23-s + (−0.951 + 0.309i)27-s + (−0.587 − 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.951 − 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.780 + 0.625i$
Analytic conductor: \(42.9859\)
Root analytic conductor: \(42.9859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (1:\ ),\ -0.780 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6590829957 + 1.876793585i\)
\(L(\frac12)\) \(\approx\) \(0.6590829957 + 1.876793585i\)
\(L(1)\) \(\approx\) \(1.136419788 + 0.5859703972i\)
\(L(1)\) \(\approx\) \(1.136419788 + 0.5859703972i\)

Euler product

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

−24.06301611425796014191781319356, −23.26392173215542787805897361849, −22.050578044884281940722732027002, −20.78171613110023603152594400638, −20.63032358395884614951632225952, −19.35670247911301037440521195712, −18.552078498640568876917636986470, −17.88008380271264947417254072042, −17.07220317026594921400076358425, −15.58043632183176842518313492195, −15.00845733597468985644670760689, −13.87330480860667848459234040829, −13.26581264930422512348857683555, −12.383766973634520175628367909364, −11.17144791518611797900919523835, −10.52932213033332241252218872857, −8.72672184697644773723111564527, −8.50314122617911573673434539003, −7.40953569749568132739446186342, −6.409227539149994664005949834384, −5.27157572962696071551739346670, −4.00091992034167989135015441605, −2.699745684298368992351367806282, −1.77208120262028143445527073189, −0.47313283487265654198998969202, 1.61945205274180900719862339667, 2.69675209388013321753493058212, 3.98342958751416423362913822880, 4.79345533167458378057558050205, 5.78120136159124075008786495548, 7.36670830774525717763835799930, 8.25667772753924291922564168498, 8.97858138229152740541231142079, 10.1482438584970648275406350065, 10.91280384175141130117758359877, 11.74745310111894839759386961866, 13.28172132680677153127621445544, 13.86043404704239194080807260622, 14.96010173823795170733571240600, 15.5384866377551708491702997955, 16.42954789885982663103643103231, 17.53072027526684036031767324179, 18.38703197902867878072083925547, 19.371902400078122943627210413525, 20.499017049095678199293680160200, 20.97889459172607760972066040111, 21.56815351954425221041427843293, 22.8276088017833366061060426958, 23.53509797895565640357348560698, 24.68562956453871279182998983124

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