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Properties

Label 1-1800-1800.877-r1-0-0
Degree $1$
Conductor $1800$
Sign $-0.995 - 0.0906i$
Analytic cond. $193.436$
Root an. cond. $193.436$
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  + (−0.866 − 0.5i)7-s + (−0.669 − 0.743i)11-s + (0.743 + 0.669i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.207 + 0.978i)23-s + (0.913 − 0.406i)29-s + (0.913 + 0.406i)31-s + (0.951 − 0.309i)37-s + (0.669 − 0.743i)41-s + (−0.866 − 0.5i)43-s + (0.406 + 0.913i)47-s + (0.5 + 0.866i)49-s + (0.587 − 0.809i)53-s + (0.669 − 0.743i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (−0.669 − 0.743i)11-s + (0.743 + 0.669i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.207 + 0.978i)23-s + (0.913 − 0.406i)29-s + (0.913 + 0.406i)31-s + (0.951 − 0.309i)37-s + (0.669 − 0.743i)41-s + (−0.866 − 0.5i)43-s + (0.406 + 0.913i)47-s + (0.5 + 0.866i)49-s + (0.587 − 0.809i)53-s + (0.669 − 0.743i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.995 - 0.0906i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ -0.995 - 0.0906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01954411648 - 0.4303948655i\)
\(L(\frac12)\) \(\approx\) \(0.01954411648 - 0.4303948655i\)
\(L(1)\) \(\approx\) \(0.8443282126 - 0.1092748091i\)
\(L(1)\) \(\approx\) \(0.8443282126 - 0.1092748091i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.207 + 0.978i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.406 + 0.913i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.994 - 0.104i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.406 + 0.913i)T \)
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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

−20.20291211735061539434192039930, −19.74101514829486151327381886847, −18.84051813577326476629629640817, −18.19565451938923020875813732098, −17.53334303939348217520751539453, −16.62958844689837160558147822513, −15.83848484434147483863933181586, −15.24450460665773430204992265238, −14.7085408520223406319751784607, −13.22089708372808712872104228607, −13.14885256706606955517204698308, −12.32383552585318913287584838636, −11.38730280916127553317334315491, −10.353521895891161268679360083865, −10.0975493457154491242580883286, −8.79659337645305968381931096971, −8.47355921930554131187589845564, −7.354694820015039311275832565476, −6.409541497145069983782308344077, −5.9672850351893713441334223555, −4.79671097596773802109805427850, −4.09591665210264757026558754338, −2.85538361515234557587168533219, −2.423784760152355424533905635563, −1.04175694204325126955396043729, 0.09375293074193911911488198953, 1.001974254786412383150179582212, 2.27340276014796736970002982594, 3.18322422455935343673445566405, 3.951146438751488600231543958511, 4.84648453713641555088526848304, 5.979370635387968200516545504843, 6.491626465932771299217255399482, 7.390665240455278191156351001080, 8.29383368356684639703723545953, 9.05994647711542753545210596616, 9.86462887520439180813293088572, 10.66213832830328771351052597204, 11.31379148966652035074260319051, 12.201893893673465399952272764020, 13.17554618261431646943417323666, 13.60332563076082163369986215178, 14.26248812298260420966616335768, 15.51094412508129091774853953365, 15.96036522438241311607521253821, 16.58180055784531762060375121674, 17.42660427381313639134072299125, 18.257556790619033421643007069526, 19.08342215843958000241625308259, 19.445120600611146758634563841027

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