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Properties

Label 1-20e2-400.147-r0-0-0
Degree $1$
Conductor $400$
Sign $0.999 + 0.0157i$
Analytic cond. $1.85759$
Root an. cond. $1.85759$
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 + i·7-s + (−0.809 + 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.809 + 0.587i)13-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.587 + 0.809i)23-s + (0.809 + 0.587i)27-s + (0.951 − 0.309i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.809 + 0.587i)37-s + (0.809 + 0.587i)39-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + i·7-s + (−0.809 + 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.809 + 0.587i)13-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.587 + 0.809i)23-s + (0.809 + 0.587i)27-s + (0.951 − 0.309i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.809 + 0.587i)37-s + (0.809 + 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.999 + 0.0157i$
Analytic conductor: \(1.85759\)
Root analytic conductor: \(1.85759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (0:\ ),\ 0.999 + 0.0157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.136078911 + 0.008922926378i\)
\(L(\frac12)\) \(\approx\) \(1.136078911 + 0.008922926378i\)
\(L(1)\) \(\approx\) \(0.9768701080 - 0.09895212930i\)
\(L(1)\) \(\approx\) \(0.9768701080 - 0.09895212930i\)

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 \( 1 + iT \)
11 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (0.951 - 0.309i)T \)
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 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−24.3211504660576191398675137616, −23.24566376072263695869493748552, −22.601529527030940817093949125059, −21.97064752291727255676541333526, −20.69204422025343212286766051972, −20.30008059308930102156001105408, −19.42597840230197481727638560929, −17.91442369082763510742860371495, −17.30040500078073352736925134727, −16.49047786383623638105580128095, −15.6881253307234057046472101019, −14.547986813004004089267113681397, −14.10740015261864295421748649823, −12.59443049075474995357128653675, −11.82014096389880047344563304589, −10.691535573244659271438941197579, −9.97128816168160722364888352446, −9.31486357271319411794887261597, −7.85689033947892331757378994175, −6.98208847272219312882857789163, −5.692103289433344634151293059956, −4.682500178119419540763136135758, −3.90304685602603240140741281897, −2.72754690944329687207766587822, −0.8400904611295053693846529714, 1.1987067019423325187594784852, 2.33695963611779756442583373450, 3.483926525651280517241129425875, 5.21422217063902189352730047554, 5.87573406541043013519554764792, 6.89379288533061657888794227829, 7.91905966430543574101265122438, 8.80996191267701106519279915982, 9.858165611952488704953450710911, 11.25482702866065938283939115312, 12.05567936412630710640766624164, 12.4569379069410714411307643518, 13.950845715904895016389661182562, 14.27251324090824549535865505236, 15.68991288068941861751153505089, 16.59328800207112261607096208759, 17.48599335123314963481994274932, 18.33330476444798668681570727735, 19.211921651175252490539428576631, 19.58487706986345940535438985117, 21.074457981947798229594366581356, 21.934066513168919185380747358945, 22.587515705734147945717535550752, 23.74469499675093229535982540494, 24.36019345751293212222786803362

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