Properties

Label 2-4800-5.4-c1-0-59
Degree $2$
Conductor $4800$
Sign $-0.894 + 0.447i$
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 − 9-s + 4·11-s − 6i·13-s + 6i·17-s − 4·19-s + i·27-s − 2·29-s − 8·31-s − 4i·33-s − 2i·37-s − 6·39-s − 6·41-s − 12i·43-s − 8i·47-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.333·9-s + 1.20·11-s − 1.66i·13-s + 1.45i·17-s − 0.917·19-s + 0.192i·27-s − 0.371·29-s − 1.43·31-s − 0.696i·33-s − 0.328i·37-s − 0.960·39-s − 0.937·41-s − 1.82i·43-s − 1.16i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 4800,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.079077209\)
\(L(\frac12)\) \(\approx\) \(1.079077209\)
\(L(\frac{3}{2})\) 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 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.088078085062497659881020821369, −7.16042493093439203673160904572, −6.61548065550678293916053145047, −5.76204000281470405600526617975, −5.31309530311752212479735715925, −3.89857759123985928834040086071, −3.60445435903543290321774038593, −2.29414568521590372842859477024, −1.50794429003346791380644018311, −0.28990047752366773298050040821, 1.35172418208740981986415162107, 2.33968411710848432369505541757, 3.40882284369630044140144217101, 4.24464911594140542575326212724, 4.65328193659855710671160174356, 5.66910998952363233241607107513, 6.55491585043535998823067326448, 6.95733613986660732011496397802, 7.88172147093439417105074732699, 8.918197958112961084216394547938

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