Properties

Label 1-200-200.123-r0-0-0
Degree $1$
Conductor $200$
Sign $0.684 - 0.728i$
Analytic cond. $0.928796$
Root an. cond. $0.928796$
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 i·7-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)13-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.951 + 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)3-s i·7-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)13-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.951 + 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.684 - 0.728i$
Analytic conductor: \(0.928796\)
Root analytic conductor: \(0.928796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (0:\ ),\ 0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327682415 - 0.5745394857i\)
\(L(\frac12)\) \(\approx\) \(1.327682415 - 0.5745394857i\)
\(L(1)\) \(\approx\) \(1.248155800 - 0.3242805703i\)
\(L(1)\) \(\approx\) \(1.248155800 - 0.3242805703i\)

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

−27.047948562069520105780792112192, −25.98935145268416485410506456059, −25.50599152674223927975625556485, −24.24206676758245369065108651699, −23.00597953611929736004173889543, −22.43001135013575351664336986364, −20.99870020097720795780757882345, −20.507560386555340266104624648855, −19.70060012943226893970477431605, −18.46320545862450584504111845155, −17.25644083241733637027345243336, −16.2674754650757809850045995568, −15.54176065990060839645011271404, −14.10191372905668073657746259611, −13.89811808352816342003662764153, −12.29891901633936347478656032970, −11.02228197847996483105985777604, −10.06250474482479072987748094494, −9.28776633679493697608595144277, −7.97993755478958190832438737693, −7.030874433402290154292534219338, −5.380111751809857909163306449695, −4.16433421536977530274035891398, −3.38210537357619214229291301878, −1.663824548384778397172246840463, 1.27602448912838674014834969047, 2.67114654466296244269418856288, 3.70851430659581806328897034640, 5.69830701228057917901575452330, 6.37057928080190450799778847587, 7.946394048040783555167129590964, 8.56280466287258145093048656492, 9.63099356663590350260381135114, 11.25853317522945616244219098300, 12.11483600912327968825894950085, 13.17630267441952030404750195040, 14.01050325542393040532471425197, 15.07166924571179995885796108190, 16.01105696324983190066011689040, 17.37971631987862278176805602138, 18.46256308766382323669849622499, 18.94496931652766911193762916932, 20.005278689639437348649542811240, 21.03408527947478219870398569794, 21.98388420966712284711947533046, 23.12851803886739554597121113980, 24.28485596696979118398664186336, 24.718876069243617777593256370191, 25.82784600403345562231676352776, 26.45738831113117726735993463129

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