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Properties

Label 1-20e2-400.373-r1-0-0
Degree $1$
Conductor $400$
Sign $-0.625 + 0.780i$
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  + (−0.809 − 0.587i)3-s − i·7-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (−0.587 − 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (0.587 − 0.809i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.309 + 0.951i)37-s + (0.309 − 0.951i)39-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s − i·7-s + (0.309 + 0.951i)9-s + (0.951 + 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (−0.587 − 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (0.587 − 0.809i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.309 + 0.951i)37-s + (0.309 − 0.951i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.625 + 0.780i$
Analytic conductor: \(42.9859\)
Root analytic conductor: \(42.9859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (1:\ ),\ -0.625 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3588643818 + 0.7473333890i\)
\(L(\frac12)\) \(\approx\) \(0.3588643818 + 0.7473333890i\)
\(L(1)\) \(\approx\) \(0.7856313849 + 0.1125083340i\)
\(L(1)\) \(\approx\) \(0.7856313849 + 0.1125083340i\)

Euler product

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

−23.51464678904990569073306009759, −23.048803968440759281768282121617, −22.214495999580710043206595353769, −21.34320492048691045424998089421, −20.39017604991886979762976652861, −19.746562856492192327930659172156, −18.34288774986998245810248427886, −17.66535497560272253589579885107, −16.51732167650332126662674214830, −16.407222279289938179579125528998, −15.004349970517325119250974518595, −14.22713449820678240666729899024, −13.100063988610412290111531784064, −12.04996845858665499861556752176, −11.22096409953718133502608749400, −10.31150813833634115126921437838, −9.67605670347085030354285620861, −8.36019728896607219297995460385, −7.18066054870921340936772921134, −6.19733564389804353515267147019, −5.276601680560133264251020024349, −4.06662691953932905957058934805, −3.41111255462304223194479396804, −1.3421605137041223966137677411, −0.27430949057509223206660820035, 1.3930553529714788118713436426, 2.284778960104088054374342324779, 3.98378826275859768627848637539, 5.0779836397156990466830460680, 6.23048016303590315781064654799, 6.67945798009770583772602257339, 8.06809697210372730332943553912, 8.985654483208648480076163037985, 10.11665034918548887151618043249, 11.30462937032748773337152999563, 11.976619662062817169343794845753, 12.62984308680437188126117293577, 13.7610191521411039478474578572, 14.7464764719763722529804961073, 15.78406239784748828068272351660, 16.74163902869623317532894507968, 17.45459504308871998428267641083, 18.40900314164182424531954473888, 19.08122235461441922655689879475, 19.87562553595370072990912226505, 21.41377010330942480084667822595, 21.8513515750096028298029712309, 22.74278840092455325735419715633, 23.72475654832580836146739284744, 24.30406434409701715004000761349

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