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Properties

Label 2-2400-40.29-c1-0-14
Degree $2$
Conductor $2400$
Sign $0.844 - 0.536i$
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  + 3-s − 3.62i·7-s + 9-s + 6.20i·11-s − 0.578·13-s − 1.42i·17-s + 5.62i·19-s − 3.62i·21-s + 5.62i·23-s + 27-s + 2i·29-s + 2.57·31-s + 6.20i·33-s + 7.83·37-s − 0.578·39-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.37i·7-s + 0.333·9-s + 1.87i·11-s − 0.160·13-s − 0.344i·17-s + 1.29i·19-s − 0.791i·21-s + 1.17i·23-s + 0.192·27-s + 0.371i·29-s + 0.463·31-s + 1.08i·33-s + 1.28·37-s − 0.0926·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.844 - 0.536i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.844 - 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.158127904\)
\(L(\frac12)\) \(\approx\) \(2.158127904\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + 3.62iT - 7T^{2} \)
11 \( 1 - 6.20iT - 11T^{2} \)
13 \( 1 + 0.578T + 13T^{2} \)
17 \( 1 + 1.42iT - 17T^{2} \)
19 \( 1 - 5.62iT - 19T^{2} \)
23 \( 1 - 5.62iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 - 7.83T + 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 - 7.25T + 43T^{2} \)
47 \( 1 - 6.78iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 2.20iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 5.42T + 79T^{2} \)
83 \( 1 + 3.25T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 4.84iT - 97T^{2} \)
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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

−9.305219737188375671908838235472, −7.908961057320675416491354488033, −7.59548461606275559727688895871, −7.01900661615058768752853171547, −6.01594146974633687935767483336, −4.75548612646811221649308172778, −4.23828884846416023973063054672, −3.39856518645524919302969834009, −2.17524481150253556806387924748, −1.20203360544240795621066385997, 0.75908279633423591072970556629, 2.47695400933385392709423147985, 2.78850816464886513641144076088, 3.95006113287529293088429752837, 4.97296994324362601039846654172, 5.92109633991245100795150168486, 6.34920568962474530654150672792, 7.53032997149301490851033767108, 8.429393469143367015833128478293, 8.766228733855145955382062037823

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