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Properties

Label 2-1800-40.13-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.772265496\)
\(L(\frac12)\) \(\approx\) \(1.772265496\)
\(L(1)\) 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

−9.590823972458598816218391845690, −8.539631843879044782965765458607, −7.83371924798380629531989275380, −7.22274378583398214906838951991, −6.60424128555511810470876836844, −5.47740960252324958243663073236, −4.49913603736955025405778657181, −4.33701070402582716012451235259, −2.97085972300348218395614389696, −1.69108136230916245936672200532, 1.25857026903807500460493415343, 2.45130827950648851578163056609, 3.23167243811718331006799878602, 4.36871636935297624993249702586, 5.19778009045072591400710342222, 5.83666076838975543748442152919, 6.58048612104266458918414747684, 7.929705517516424324645840620353, 8.655251282167984651836593106146, 9.238758406862070055022440777792

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