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Properties

Label 2-2400-20.3-c1-0-15
Degree $2$
Conductor $2400$
Sign $0.973 + 0.229i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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  + (−0.707 + 0.707i)3-s + (−0.585 − 0.585i)7-s − 1.00i·9-s + 0.828i·11-s + (−1.41 − 1.41i)13-s + (−0.828 + 0.828i)17-s + 3.65·19-s + 0.828·21-s + (−2 + 2i)23-s + (0.707 + 0.707i)27-s − 5.65i·29-s + 6i·31-s + (−0.585 − 0.585i)33-s + (5.41 − 5.41i)37-s + 2.00·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.221 − 0.221i)7-s − 0.333i·9-s + 0.249i·11-s + (−0.392 − 0.392i)13-s + (−0.200 + 0.200i)17-s + 0.838·19-s + 0.180·21-s + (−0.417 + 0.417i)23-s + (0.136 + 0.136i)27-s − 1.05i·29-s + 1.07i·31-s + (−0.101 − 0.101i)33-s + (0.890 − 0.890i)37-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.321696587\)
\(L(\frac12)\) \(\approx\) \(1.321696587\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (0.585 + 0.585i)T + 7iT^{2} \)
11 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 + (0.828 - 0.828i)T - 17iT^{2} \)
19 \( 1 - 3.65T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + (-5.41 + 5.41i)T - 37iT^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (0.828 + 0.828i)T + 47iT^{2} \)
53 \( 1 + (6.82 + 6.82i)T + 53iT^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (6.82 + 6.82i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (-8.48 - 8.48i)T + 73iT^{2} \)
79 \( 1 - 9.31T + 79T^{2} \)
83 \( 1 + (1.17 - 1.17i)T - 83iT^{2} \)
89 \( 1 + 11.6iT - 89T^{2} \)
97 \( 1 + (-1.17 + 1.17i)T - 97iT^{2} \)
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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

−9.058151550653546733706617346209, −8.124075273106751566705952825548, −7.37860480056919113596070539855, −6.59474943149775195497250857328, −5.71285962437596355843086014127, −5.03772767690865269289950168294, −4.10813621786364873053736092686, −3.30604938802775345986726930135, −2.13257807370486007222680568383, −0.63882784426825828617881436044, 0.878093604955366754523195541136, 2.17486678879748302912785047399, 3.11291147221046229336346407371, 4.27740484933396247978511086524, 5.11526953904957063581136506049, 5.96537982709506937944249442355, 6.61218632788694044545240848859, 7.45490837470988208550804550395, 8.086952237218480514312100495313, 9.081723279228659751915725433883

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