Properties

Label 2-8048-1.1-c1-0-130
Degree $2$
Conductor $8048$
Sign $1$
Analytic cond. $64.2636$
Root an. cond. $8.01645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.46·3-s − 4.08·5-s + 4.62·7-s + 3.07·9-s + 4.22·11-s + 4.60·13-s − 10.0·15-s + 0.221·17-s + 0.311·19-s + 11.4·21-s + 7.93·23-s + 11.6·25-s + 0.175·27-s − 4.36·29-s − 3.02·31-s + 10.4·33-s − 18.8·35-s + 7.63·37-s + 11.3·39-s − 8.18·41-s + 4.18·43-s − 12.5·45-s − 2.06·47-s + 14.4·49-s + 0.544·51-s + 10.6·53-s − 17.2·55-s + ⋯
L(s)  = 1  + 1.42·3-s − 1.82·5-s + 1.74·7-s + 1.02·9-s + 1.27·11-s + 1.27·13-s − 2.59·15-s + 0.0536·17-s + 0.0714·19-s + 2.48·21-s + 1.65·23-s + 2.33·25-s + 0.0338·27-s − 0.810·29-s − 0.543·31-s + 1.81·33-s − 3.19·35-s + 1.25·37-s + 1.81·39-s − 1.27·41-s + 0.637·43-s − 1.86·45-s − 0.300·47-s + 2.05·49-s + 0.0762·51-s + 1.46·53-s − 2.32·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8048\)    =    \(2^{4} \cdot 503\)
Sign: $1$
Analytic conductor: \(64.2636\)
Root analytic conductor: \(8.01645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8048,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.913260936\)
\(L(\frac12)\) \(\approx\) \(3.913260936\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 - T \)
good3 \( 1 - 2.46T + 3T^{2} \)
5 \( 1 + 4.08T + 5T^{2} \)
7 \( 1 - 4.62T + 7T^{2} \)
11 \( 1 - 4.22T + 11T^{2} \)
13 \( 1 - 4.60T + 13T^{2} \)
17 \( 1 - 0.221T + 17T^{2} \)
19 \( 1 - 0.311T + 19T^{2} \)
23 \( 1 - 7.93T + 23T^{2} \)
29 \( 1 + 4.36T + 29T^{2} \)
31 \( 1 + 3.02T + 31T^{2} \)
37 \( 1 - 7.63T + 37T^{2} \)
41 \( 1 + 8.18T + 41T^{2} \)
43 \( 1 - 4.18T + 43T^{2} \)
47 \( 1 + 2.06T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 9.22T + 59T^{2} \)
61 \( 1 - 8.58T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 + 0.575T + 73T^{2} \)
79 \( 1 + 3.32T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−7.927195293953982499252293237957, −7.39387266739446666285270429365, −6.90018392833249954241290482166, −5.64187600404075610250246651700, −4.60666568733089198879266661415, −4.12508880419517368354937362097, −3.60290412557697572678890595629, −2.90783384555505129974646912564, −1.65837044227591970889694497498, −1.03155805993454447028738255362, 1.03155805993454447028738255362, 1.65837044227591970889694497498, 2.90783384555505129974646912564, 3.60290412557697572678890595629, 4.12508880419517368354937362097, 4.60666568733089198879266661415, 5.64187600404075610250246651700, 6.90018392833249954241290482166, 7.39387266739446666285270429365, 7.927195293953982499252293237957

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