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Properties

Label 2-1800-360.203-c0-0-4
Degree $2$
Conductor $1800$
Sign $0.548 - 0.835i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
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  + (−0.258 + 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 − 0.499i)4-s + (−0.499 + 0.866i)6-s + (0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (1.5 − 0.866i)11-s + (−0.707 − 0.707i)12-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)17-s + (−0.707 + 0.707i)18-s − i·19-s + (0.448 + 1.67i)22-s + (0.866 − 0.500i)24-s + (0.707 + 0.707i)27-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 − 0.499i)4-s + (−0.499 + 0.866i)6-s + (0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (1.5 − 0.866i)11-s + (−0.707 − 0.707i)12-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)17-s + (−0.707 + 0.707i)18-s − i·19-s + (0.448 + 1.67i)22-s + (0.866 − 0.500i)24-s + (0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.548 - 0.835i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.548 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.386839858\)
\(L(\frac12)\) \(\approx\) \(1.386839858\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.243657012177352001130899638379, −8.803142301681096997356143400834, −8.209091185308231752000742215823, −7.08249693596796219427343193299, −6.75232378927737669149813895298, −5.65346556905527983662545161732, −4.59202264730310450612542447013, −3.94740464080950153980125758505, −2.85379921839343899190568397783, −1.30115876806436985692696945312, 1.53123519881036319267111811396, 2.09151935841214213745369723556, 3.48048628370840152844591991911, 3.94213685273469151199995608222, 4.86951811138171394480055467653, 6.38017300186190150669460747550, 7.12413222456726053451812916106, 8.106155278093273670144797280002, 8.662971551442719217954940423188, 9.408661376828354589555055463553

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