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Properties

Label 2-1800-15.8-c1-0-10
Degree $2$
Conductor $1800$
Sign $0.662 + 0.749i$
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  + (−1.23 + 1.23i)7-s + 1.74i·11-s + (−0.236 − 0.236i)13-s + (−4.57 − 4.57i)17-s − 6.47i·19-s + (2.82 − 2.82i)23-s + 0.333·29-s + 10.4·31-s + (2.23 − 2.23i)37-s − 7.07i·41-s + (6.47 + 6.47i)43-s + (4.57 + 4.57i)47-s + 3.94i·49-s − 7.40·59-s + 1.52·61-s + ⋯
L(s)  = 1  + (−0.467 + 0.467i)7-s + 0.527i·11-s + (−0.0654 − 0.0654i)13-s + (−1.10 − 1.10i)17-s − 1.48i·19-s + (0.589 − 0.589i)23-s + 0.0619·29-s + 1.88·31-s + (0.367 − 0.367i)37-s − 1.10i·41-s + (0.986 + 0.986i)43-s + (0.667 + 0.667i)47-s + 0.563i·49-s − 0.964·59-s + 0.195·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.391972088\)
\(L(\frac12)\) \(\approx\) \(1.391972088\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.23 - 1.23i)T - 7iT^{2} \)
11 \( 1 - 1.74iT - 11T^{2} \)
13 \( 1 + (0.236 + 0.236i)T + 13iT^{2} \)
17 \( 1 + (4.57 + 4.57i)T + 17iT^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 0.333T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (-2.23 + 2.23i)T - 37iT^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (-6.47 - 6.47i)T + 43iT^{2} \)
47 \( 1 + (-4.57 - 4.57i)T + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 7.40T + 59T^{2} \)
61 \( 1 - 1.52T + 61T^{2} \)
67 \( 1 + (-10.4 + 10.4i)T - 67iT^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + (-9.47 - 9.47i)T + 73iT^{2} \)
79 \( 1 + 5.52iT - 79T^{2} \)
83 \( 1 + (-7.40 + 7.40i)T - 83iT^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (1 - i)T - 97iT^{2} \)
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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

−9.262605123423143206492378571876, −8.542339741980555390969124247681, −7.46385480118720303189234488552, −6.78459928321386821922515516665, −6.09729202744039035080820933543, −4.87829349696467192440395893545, −4.44566450367066314875533661296, −2.91733790143508744891107791241, −2.39131648235953259899952343647, −0.60031373427934393251865857819, 1.10637291333502931920810480645, 2.44759269272755651224474711070, 3.61403818202462773374309798522, 4.24111250223019309977256458591, 5.42467432193379890917798214636, 6.28914686662444448692166392114, 6.84916064602180578077827475531, 7.982690521293566277973405479549, 8.458978314032344454386948263157, 9.440012704636500000132725270361

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