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Properties

Label 1-1800-1800.283-r0-0-0
Degree $1$
Conductor $1800$
Sign $-0.282 - 0.959i$
Analytic cond. $8.35916$
Root an. cond. $8.35916$
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 + (−0.104 − 0.994i)11-s + (0.994 + 0.104i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.406 − 0.913i)23-s + (0.669 + 0.743i)29-s + (−0.669 + 0.743i)31-s + (0.587 − 0.809i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + (−0.743 + 0.669i)47-s + (0.5 + 0.866i)49-s + (−0.951 − 0.309i)53-s + (0.104 − 0.994i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (−0.104 − 0.994i)11-s + (0.994 + 0.104i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.406 − 0.913i)23-s + (0.669 + 0.743i)29-s + (−0.669 + 0.743i)31-s + (0.587 − 0.809i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + (−0.743 + 0.669i)47-s + (0.5 + 0.866i)49-s + (−0.951 − 0.309i)53-s + (0.104 − 0.994i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.282 - 0.959i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ -0.282 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7154128132 - 0.9563120453i\)
\(L(\frac12)\) \(\approx\) \(0.7154128132 - 0.9563120453i\)
\(L(1)\) \(\approx\) \(0.9410472772 - 0.2536888798i\)
\(L(1)\) \(\approx\) \(0.9410472772 - 0.2536888798i\)

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 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (0.406 - 0.913i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.743 + 0.669i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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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

−20.486432079928769715138425224827, −19.60950489417133291476179747751, −18.86568573074349023896914041308, −18.38216305759176323555917614308, −17.461054857783783180520400978120, −16.67709935122488227852054913886, −15.989885414933240631694515960682, −15.21889099742006059431014363875, −14.6954733150018975810953561020, −13.54523919933713978471012487016, −13.003568359073804149681468043193, −12.21867714292084285411759511657, −11.61021716115775252908817660162, −10.46454457426347549177578244972, −9.88258266568105017374576692685, −9.19496643260135824093984051009, −8.21864264103395162054662248879, −7.53297508319854988957939689004, −6.45994174676887924254516352804, −5.919218886839176855590665102562, −5.02727244471495524087920725797, −3.86967421481268620289475851, −3.28147535577269047304142742344, −2.17771480864276585083531719090, −1.24068654944123466582126689229, 0.46138034949598045347684350471, 1.41654784267195725434278253375, 2.96869641646364166519646932571, 3.29259612560864777304468124232, 4.37072398052307185994252786526, 5.33990819408645017064282597067, 6.30115197924373607759938668774, 6.77952246535715244986235116581, 7.82322316670834672026455295279, 8.69352640555616012067267821884, 9.30020914275797933776632094604, 10.34801393007774099140825096967, 10.86947285034478141882001357484, 11.66707708166799772674035691547, 12.76629462964260993078082160995, 13.191033674084316431516091675933, 14.03143270931911209023178943992, 14.6615134513678450160367511079, 15.86171780577633818467936651551, 16.26087061425857091675787736448, 16.807852418888443782667669893119, 17.88392275109788566203080702899, 18.57548818759004360059786289531, 19.25030860105040806514918441042, 19.902336621141391309053686784916

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