Properties

Label 2-1800-360.203-c0-0-2
Degree $2$
Conductor $1800$
Sign $-0.485 - 0.874i$
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 + (−0.707 + 0.707i)3-s + (−0.866 − 0.499i)4-s + (−0.500 − 0.866i)6-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.965 − 0.258i)12-s + (0.500 + 0.866i)16-s + (1.41 + 1.41i)17-s + (0.965 + 0.258i)18-s i·19-s + i·24-s + (0.707 + 0.707i)27-s + (−0.965 + 0.258i)32-s + (−1.73 + 1.00i)34-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.707 + 0.707i)3-s + (−0.866 − 0.499i)4-s + (−0.500 − 0.866i)6-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.965 − 0.258i)12-s + (0.500 + 0.866i)16-s + (1.41 + 1.41i)17-s + (0.965 + 0.258i)18-s i·19-s + i·24-s + (0.707 + 0.707i)27-s + (−0.965 + 0.258i)32-s + (−1.73 + 1.00i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.485 - 0.874i$
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: \((2,\ 1800,\ (\ :0),\ -0.485 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7145169227\)
\(L(\frac12)\) \(\approx\) \(0.7145169227\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
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

−9.561389540031988088865156655943, −9.059709104164981330523856125334, −8.063400574693650083019136006483, −7.38658554919689517101239304191, −6.22415234123759863791052863632, −5.96186106408533983196218075924, −4.92759265151079259715679660242, −4.27342018031380039672694363437, −3.24329077490562343282740099791, −1.16423244514000774639841882485, 0.815501324561288130351982581728, 1.97360142718853277560915551834, 3.02955244935829610636045908990, 4.11353697261956627823289694751, 5.24055957727984368598175476072, 5.72629474717964748050886325106, 7.08177321246319718839540973977, 7.63584384205201546866811586109, 8.419627107563758700468507482547, 9.392513656807234442262780525609

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