Properties

Label 2-8048-1.1-c1-0-145
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.60·3-s − 1.79·5-s + 2.96·7-s + 3.80·9-s + 0.556·11-s + 3.76·13-s + 4.69·15-s + 1.41·17-s − 1.17·19-s − 7.72·21-s − 0.626·23-s − 1.76·25-s − 2.11·27-s − 0.654·29-s − 1.67·31-s − 1.45·33-s − 5.32·35-s − 2.59·37-s − 9.83·39-s + 6.68·41-s − 11.7·43-s − 6.85·45-s − 13.0·47-s + 1.76·49-s − 3.69·51-s + 9.66·53-s − 1.00·55-s + ⋯
L(s)  = 1  − 1.50·3-s − 0.804·5-s + 1.11·7-s + 1.26·9-s + 0.167·11-s + 1.04·13-s + 1.21·15-s + 0.343·17-s − 0.270·19-s − 1.68·21-s − 0.130·23-s − 0.352·25-s − 0.406·27-s − 0.121·29-s − 0.300·31-s − 0.252·33-s − 0.900·35-s − 0.426·37-s − 1.57·39-s + 1.04·41-s − 1.79·43-s − 1.02·45-s − 1.89·47-s + 0.252·49-s − 0.517·51-s + 1.32·53-s − 0.134·55-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: \(1\)
Selberg data: \((2,\ 8048,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 3T^{2} \)
5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 - 2.96T + 7T^{2} \)
11 \( 1 - 0.556T + 11T^{2} \)
13 \( 1 - 3.76T + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 + 0.626T + 23T^{2} \)
29 \( 1 + 0.654T + 29T^{2} \)
31 \( 1 + 1.67T + 31T^{2} \)
37 \( 1 + 2.59T + 37T^{2} \)
41 \( 1 - 6.68T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 - 9.66T + 53T^{2} \)
59 \( 1 + 5.84T + 59T^{2} \)
61 \( 1 + 6.37T + 61T^{2} \)
67 \( 1 + 9.80T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 2.17T + 79T^{2} \)
83 \( 1 + 2.48T + 83T^{2} \)
89 \( 1 + 1.95T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−7.48520086433505278605055756887, −6.62445835343576149931414609496, −6.11816558608838831631002156026, −5.30889902051535228256692718225, −4.82496214141750234904194461782, −4.07408460993263785537010398238, −3.38607282453465332534015713575, −1.86809443062259786302497519914, −1.08883716736600991845870874913, 0, 1.08883716736600991845870874913, 1.86809443062259786302497519914, 3.38607282453465332534015713575, 4.07408460993263785537010398238, 4.82496214141750234904194461782, 5.30889902051535228256692718225, 6.11816558608838831631002156026, 6.62445835343576149931414609496, 7.48520086433505278605055756887

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