Properties

Label 2-4800-5.4-c1-0-3
Degree $2$
Conductor $4800$
Sign $0.447 - 0.894i$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
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 − 3i·7-s − 9-s − 2·11-s + 3i·13-s − 6i·17-s − 7·19-s − 3·21-s + 6i·23-s + i·27-s − 2·29-s − 5·31-s + 2i·33-s + 10i·37-s + 3·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.13i·7-s − 0.333·9-s − 0.603·11-s + 0.832i·13-s − 1.45i·17-s − 1.60·19-s − 0.654·21-s + 1.25i·23-s + 0.192i·27-s − 0.371·29-s − 0.898·31-s + 0.348i·33-s + 1.64i·37-s + 0.480·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(38.3281\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7398453288\)
\(L(\frac12)\) \(\approx\) \(0.7398453288\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 3iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−8.251711654258807428969878703982, −7.48864256819948257202563261604, −7.13719852191543241130875719535, −6.39676063706975165436882843172, −5.53724449978106674949920861592, −4.62142884779934403994223876580, −3.99056034914336343200142618524, −2.94736955497131800590406989821, −2.03726676943916773429991317158, −0.987627823977419018255840836011, 0.21998672096031077742442897395, 2.06766597255283751538373498990, 2.57417481469800850481440410396, 3.71503505122769703031839500594, 4.35569055707554968331559678215, 5.39102918851410175269321190964, 5.81477532772943756049781928814, 6.48954271962727643542959951163, 7.61960859333694318278678758499, 8.382180679225378054065618073833

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