Properties

Label 1-20e2-400.181-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  + (−0.951 − 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

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

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: $\chi_{400} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (0:\ ),\ 0.999 + 0.0157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7497396406 + 0.005888562446i\)
\(L(\frac12)\) \(\approx\) \(0.7497396406 + 0.005888562446i\)
\(L(1)\) \(\approx\) \(0.7174299990 - 0.03784946834i\)
\(L(1)\) \(\approx\) \(0.7174299990 - 0.03784946834i\)

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

−24.29160324893042942553091699277, −23.23259291865532970096245266931, −22.60116537585744822857481165832, −22.16661872785064039754963026260, −20.84912145739060373382491885825, −20.24725780211287681218955301147, −18.97410091021990670710787850648, −18.199043201538547107271215085496, −17.31735552700927723450750570731, −16.4814722219393784551346358562, −15.6591167880414549849762419682, −14.99985638825576340906719766240, −13.47640760266867918905583364015, −12.66668499531369840496596139178, −11.9396995620045842449600288152, −10.861324027320343009910366814076, −9.81225944034377699621612782433, −9.49672281524638383844833574575, −7.60527751735034078742413752785, −6.97479197652037548843726626602, −5.663246332178225912903301797272, −5.089549262425852212742311131879, −3.75872511133070167882466850912, −2.61495824419728581285073740853, −0.74359523408849577420852551530, 0.84535873166663269741674491860, 2.44150402651114286368472203672, 3.73207230695164471972822147880, 5.03455790489786951954911020830, 5.947185205159160629527675636086, 6.78434880216272869866469430953, 7.713173535351097347615610787718, 9.08453320973885711077090844244, 10.08054793490991510452940532000, 10.95138778706143693008866288017, 11.87074737132311725017006270301, 12.815753709107907963271609887783, 13.40892119478809843809576271323, 14.66378677821605459801743664533, 15.96184341360770038120169158388, 16.443503804625648244849408099073, 17.249505308087375787858650407711, 18.371974465321398791893940860430, 19.03724930850063943156088894957, 19.75938365774597882321072042529, 21.26801470776570277324882489162, 21.814098949170434234658468352455, 22.73941710087228679431040342243, 23.440576193594804064124039866597, 24.289543228645935775978841968654

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