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Properties

Label 1-200-200.69-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  + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)33-s + (−0.809 − 0.587i)37-s + (−0.809 + 0.587i)39-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)33-s + (−0.809 − 0.587i)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: $\chi_{200} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (0:\ ),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4606798849 - 0.8379735999i\)
\(L(\frac12)\) \(\approx\) \(0.4606798849 - 0.8379735999i\)
\(L(1)\) \(\approx\) \(0.8380630645 - 0.4769246936i\)
\(L(1)\) \(\approx\) \(0.8380630645 - 0.4769246936i\)

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

−27.146581540236877263392167100801, −26.274107403128604008315849999692, −25.536893080085484107015865310849, −24.61716559063018318793444766447, −23.1477543843218748245200411491, −22.39542309154696727405600154112, −21.62796812648621775685501032715, −20.59037390961323736065557425051, −19.57238688653047279060108563735, −19.02736818006370401812132181031, −17.11462317916465659479797723837, −16.810480442397280437383458710205, −15.4593829923634562876744817420, −14.84674485925922985049363345284, −13.72016513383307300127703926636, −12.51991374006951854849347439082, −11.42107835600971182950707286983, −10.046575400527769110662069345502, −9.5714032027999046730488842809, −8.45037151115983159539947776757, −6.99095350482237177472513450693, −5.82691498520765418851432696321, −4.38853032038681970214765221973, −3.568778187344803239033126490217, −2.145264895084835188911542033013, 0.69389153460609489405336817747, 2.49663684854194352887805900729, 3.412165703772744130978189889159, 5.22682935438834922934244502642, 6.62480938226212239112818770165, 7.133996391907092231978195959120, 8.655835014698588588953635930061, 9.37977390434927547472105245954, 10.87049136221029491914370742198, 12.0976594884517012309736627881, 12.86663337868514326116248969783, 13.79744953967671234304667022997, 14.75148520223074153234419685416, 15.998350023285272977335228645253, 17.099949634553803972934306936004, 17.99089897364909189729328011408, 19.24155789376734856979275753314, 19.55966020520372132120357428345, 20.6390720631732519244668956466, 22.160553711729395439083705616515, 22.719833212164619999131675167299, 23.92636190492385398930530567650, 24.77152412220844144464136308052, 25.44391922050916595312267324178, 26.45598364628349757858082687017

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