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Properties

Label 2-1800-40.13-c0-0-2
Degree $2$
Conductor $1800$
Sign $-0.229 + 0.973i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (0.707 − 0.707i)8-s − 1.41i·11-s − 1.41·14-s − 1.00·16-s + (−1.00 + 1.00i)22-s + (1.00 + 1.00i)28-s − 1.41·29-s + (0.707 + 0.707i)32-s + 1.41·44-s − i·49-s − 1.41i·56-s + (1.00 + 1.00i)58-s + 1.41·59-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (0.707 − 0.707i)8-s − 1.41i·11-s − 1.41·14-s − 1.00·16-s + (−1.00 + 1.00i)22-s + (1.00 + 1.00i)28-s − 1.41·29-s + (0.707 + 0.707i)32-s + 1.41·44-s − i·49-s − 1.41i·56-s + (1.00 + 1.00i)58-s + 1.41·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8403258287\)
\(L(\frac12)\) \(\approx\) \(0.8403258287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.168752827918586080242898225086, −8.533989473328840698710183291250, −7.79138821621787615637513895828, −7.25279458696588505105816610914, −6.11542491827467045843340964952, −4.97824090418571031361734352697, −3.98755389708202670470417382156, −3.28837668714511074863237829324, −1.97089657785380041480085427782, −0.849184209890840376901927111587, 1.64056384781367897080050106323, 2.36646873617890536542613806966, 4.18481187228071016322039162884, 5.10219579332562101002688585285, 5.61301001469599497995527214091, 6.69106783609463915421471842486, 7.44678836541089832807479264207, 8.093391015137002666452359742275, 8.871996823880700707533906084800, 9.499141390016413142562050026815

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