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Properties

Label 2-1800-120.59-c1-0-17
Degree $2$
Conductor $1800$
Sign $0.472 - 0.881i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 2.00·4-s − 4.24·7-s − 2.82·8-s + 1.41i·11-s + 4.24·13-s + 6·14-s + 4.00·16-s − 2.82·17-s + 4·19-s − 2.00i·22-s − 6i·23-s − 6·26-s − 8.48·28-s − 6·29-s + ⋯
L(s)  = 1  − 1.00·2-s + 1.00·4-s − 1.60·7-s − 1.00·8-s + 0.426i·11-s + 1.17·13-s + 1.60·14-s + 1.00·16-s − 0.685·17-s + 0.917·19-s − 0.426i·22-s − 1.25i·23-s − 1.17·26-s − 1.60·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.472 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6999510961\)
\(L(\frac12)\) \(\approx\) \(0.6999510961\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 9.89iT - 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.584365655482026437905669080863, −8.773132129961797696657834429046, −7.900487445874186740400429411427, −7.07675775397074098470606800079, −6.27669763922342834423458663734, −5.90314517676091538970993842284, −4.26477595082095579210338214886, −3.24643058733835535148939869315, −2.41820618665762689970894847876, −0.929565775975984558024639647398, 0.46133404999719487496361745653, 1.85422973352389487854389971886, 3.29256626011906059650741348982, 3.56478487847955113820716990140, 5.46804234497602900848665035910, 6.11775819594943899220946536917, 6.88751795580207758512883849265, 7.50336338689315329534488208149, 8.670228878992744566011724710795, 9.075858613142019848020235983184

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