Properties

Label 2-2400-5.4-c1-0-20
Degree $2$
Conductor $2400$
Sign $0.447 + 0.894i$
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  i·3-s − 9-s − 4·11-s + 2i·13-s + 2i·17-s + 8·19-s − 4i·23-s + i·27-s + 6·29-s + 4i·33-s − 2i·37-s + 2·39-s − 6·41-s − 4i·43-s − 12i·47-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.333·9-s − 1.20·11-s + 0.554i·13-s + 0.485i·17-s + 1.83·19-s − 0.834i·23-s + 0.192i·27-s + 1.11·29-s + 0.696i·33-s − 0.328i·37-s + 0.320·39-s − 0.937·41-s − 0.609i·43-s − 1.75i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.559654199\)
\(L(\frac12)\) \(\approx\) \(1.559654199\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.529389122583693293741640865042, −8.190393920947441415700813720900, −7.15806364265476234137745060548, −6.75148687208535328069321928687, −5.55339513595594029290523741562, −5.12042229147671765131384252835, −3.89059680202904436367062816236, −2.87783097938211242420526184109, −1.99851166935410544626595899634, −0.65018523303646176530392063763, 1.00217395972657893711427271711, 2.68003945870910668634102494478, 3.20338781620179052333488351100, 4.36816029174412791771296539175, 5.32840170322913764121323210682, 5.58966397014236965753719663333, 6.90459188054174155988711236385, 7.66662785429589770897642491505, 8.270377436592026487822030625073, 9.201145566901437476918609800502

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