Properties

Label 2-8048-1.1-c1-0-137
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  + 2.88·3-s − 1.00·5-s + 1.34·7-s + 5.30·9-s + 0.428·11-s + 2.65·13-s − 2.88·15-s + 8.02·17-s − 4.96·19-s + 3.87·21-s + 7.95·23-s − 3.99·25-s + 6.64·27-s + 7.89·29-s + 1.02·31-s + 1.23·33-s − 1.34·35-s − 6.45·37-s + 7.63·39-s − 0.415·41-s + 2.10·43-s − 5.31·45-s + 2.82·47-s − 5.18·49-s + 23.1·51-s − 10.7·53-s − 0.429·55-s + ⋯
L(s)  = 1  + 1.66·3-s − 0.448·5-s + 0.508·7-s + 1.76·9-s + 0.129·11-s + 0.735·13-s − 0.745·15-s + 1.94·17-s − 1.13·19-s + 0.846·21-s + 1.65·23-s − 0.799·25-s + 1.27·27-s + 1.46·29-s + 0.184·31-s + 0.214·33-s − 0.227·35-s − 1.06·37-s + 1.22·39-s − 0.0649·41-s + 0.321·43-s − 0.792·45-s + 0.412·47-s − 0.741·49-s + 3.23·51-s − 1.47·53-s − 0.0578·55-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

\(L(1)\) \(\approx\) \(4.545907087\)
\(L(\frac12)\) \(\approx\) \(4.545907087\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.88T + 3T^{2} \)
5 \( 1 + 1.00T + 5T^{2} \)
7 \( 1 - 1.34T + 7T^{2} \)
11 \( 1 - 0.428T + 11T^{2} \)
13 \( 1 - 2.65T + 13T^{2} \)
17 \( 1 - 8.02T + 17T^{2} \)
19 \( 1 + 4.96T + 19T^{2} \)
23 \( 1 - 7.95T + 23T^{2} \)
29 \( 1 - 7.89T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 + 6.45T + 37T^{2} \)
41 \( 1 + 0.415T + 41T^{2} \)
43 \( 1 - 2.10T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 2.97T + 59T^{2} \)
61 \( 1 - 9.70T + 61T^{2} \)
67 \( 1 - 1.01T + 67T^{2} \)
71 \( 1 + 6.32T + 71T^{2} \)
73 \( 1 - 4.36T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 9.87T + 83T^{2} \)
89 \( 1 + 6.59T + 89T^{2} \)
97 \( 1 + 2.55T + 97T^{2} \)
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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

−8.024203020663970249560615875335, −7.37264065876116182357877316513, −6.71284106304303231248921905687, −5.73094637205078574863011338849, −4.79698622816755147724673978973, −4.08328469176137828400729898040, −3.34992989107726824360307654720, −2.90144139438214178682793455006, −1.82124111065413357038288015190, −1.06005946127687910347124524106, 1.06005946127687910347124524106, 1.82124111065413357038288015190, 2.90144139438214178682793455006, 3.34992989107726824360307654720, 4.08328469176137828400729898040, 4.79698622816755147724673978973, 5.73094637205078574863011338849, 6.71284106304303231248921905687, 7.37264065876116182357877316513, 8.024203020663970249560615875335

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