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Properties

Label 2-1800-15.14-c0-0-3
Degree $2$
Conductor $1800$
Sign $-0.151 + 0.988i$
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  − i·7-s − 1.41i·11-s − i·13-s − 1.41·17-s − 19-s − 1.41·23-s + 1.41i·29-s + 31-s − i·43-s + 1.41·47-s − 1.41i·59-s + 61-s − i·67-s − 1.41·77-s + 1.41·83-s + ⋯
L(s)  = 1  − i·7-s − 1.41i·11-s − i·13-s − 1.41·17-s − 19-s − 1.41·23-s + 1.41i·29-s + 31-s − i·43-s + 1.41·47-s − 1.41i·59-s + 61-s − i·67-s − 1.41·77-s + 1.41·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8962438985\)
\(L(\frac12)\) \(\approx\) \(0.8962438985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−9.095505588215803017095146279506, −8.429089448080077108606297538093, −7.80771135216142833950577531601, −6.76623265050925773797332647609, −6.16579211561294732878249257368, −5.18960432241956178271574140990, −4.14224504853309267422794813555, −3.44643228120741976196471363593, −2.24535248636476595986469860754, −0.64173593678860447422401944149, 2.06237253294049118076745840337, 2.39692528249385055596350221957, 4.31239316300375921860476195823, 4.40149421074976282478932902633, 5.81715931934439561235197827018, 6.45423382733856997122716715762, 7.23849063834502085417086444428, 8.259711359647712862088154165644, 8.895069389927032288525283347461, 9.641793135968397460364609789811

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