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Properties

Label 1-200-200.83-r0-0-0
Degree $1$
Conductor $200$
Sign $0.535 + 0.844i$
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  + (0.951 + 0.309i)3-s − i·7-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.587 + 0.809i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.951 + 0.309i)33-s + (0.587 − 0.809i)37-s + (−0.809 + 0.587i)39-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s − i·7-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.587 + 0.809i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.951 + 0.309i)33-s + (0.587 − 0.809i)37-s + (−0.809 + 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(0.928796\)
Root analytic conductor: \(0.928796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (0:\ ),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330437507 + 0.7314142092i\)
\(L(\frac12)\) \(\approx\) \(1.330437507 + 0.7314142092i\)
\(L(1)\) \(\approx\) \(1.301618751 + 0.3720137471i\)
\(L(1)\) \(\approx\) \(1.301618751 + 0.3720137471i\)

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

−26.93185889189180907294698077233, −25.69478187993929259586589584740, −25.057218372201281526907852093831, −23.84695358308389793125871893659, −23.30533120330761862689100286570, −21.85997820066507556020114056308, −20.76089583319993999275251674066, −20.21149632525028058632342280305, −19.12791804142954286390912164308, −18.37519083131514066993121095758, −17.10719173790124431731447556198, −16.12000491240526747016399347158, −14.855835132561706732964236497766, −14.15270260689941362030684742724, −13.11512965548114286876473234699, −12.37289117543238648538428729218, −10.597158553225062970959311192553, −10.01796944526591975294227213176, −8.47748487471459852625527952871, −7.791733326699066738017825111254, −6.76095418778843723688424577846, −5.193424260483532687942049777694, −3.723723069796121488762553762351, −2.80465940372880438118563426778, −1.164500797690574424686146980997, 2.06791314829376274656001823342, 2.888127830577544412438039938, 4.40498455688820979195841030927, 5.44849896318942906127635971978, 7.11820679673594287153428588896, 8.06452406711782498167433745646, 9.255870255293980056937233389367, 9.82906002786716369283017325983, 11.30629698701654437072885696411, 12.49474840171789441244327410161, 13.445251537726938769291721117835, 14.622531484576252527918404202943, 15.30786164225082126748149628534, 16.1657579770685404264022749728, 17.53150472669407552651181828207, 18.77260196357613596072976214332, 19.27861724351885856650081797036, 20.53141859857687723526072586041, 21.32212972887423024688155582477, 22.00651947624602412083201430630, 23.37515030046129041314211087271, 24.41368278561819095663549465435, 25.353133961191990065029031752152, 25.96281983928991448547171721798, 26.94959744336115842910097857188

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