Properties

Degree 2
Conductor $ 2^{4} \cdot 503 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.799·3-s + 0.626·5-s + 0.555·7-s − 2.36·9-s + 1.34·11-s + 2.90·13-s − 0.500·15-s + 2.10·17-s + 8.54·19-s − 0.444·21-s − 8.01·23-s − 4.60·25-s + 4.28·27-s − 2.10·29-s − 3.16·31-s − 1.07·33-s + 0.347·35-s − 10.9·37-s − 2.31·39-s − 0.458·41-s − 2.58·43-s − 1.47·45-s − 10.5·47-s − 6.69·49-s − 1.68·51-s − 5.54·53-s + 0.841·55-s + ⋯
L(s)  = 1  − 0.461·3-s + 0.279·5-s + 0.209·7-s − 0.787·9-s + 0.405·11-s + 0.804·13-s − 0.129·15-s + 0.510·17-s + 1.96·19-s − 0.0969·21-s − 1.67·23-s − 0.921·25-s + 0.824·27-s − 0.390·29-s − 0.568·31-s − 0.186·33-s + 0.0587·35-s − 1.80·37-s − 0.371·39-s − 0.0716·41-s − 0.394·43-s − 0.220·45-s − 1.53·47-s − 0.955·49-s − 0.235·51-s − 0.761·53-s + 0.113·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8048\)    =    \(2^{4} \cdot 503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8048} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8048,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
503 \( 1 + T \)
good3 \( 1 + 0.799T + 3T^{2} \)
5 \( 1 - 0.626T + 5T^{2} \)
7 \( 1 - 0.555T + 7T^{2} \)
11 \( 1 - 1.34T + 11T^{2} \)
13 \( 1 - 2.90T + 13T^{2} \)
17 \( 1 - 2.10T + 17T^{2} \)
19 \( 1 - 8.54T + 19T^{2} \)
23 \( 1 + 8.01T + 23T^{2} \)
29 \( 1 + 2.10T + 29T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 0.458T + 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 5.54T + 53T^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 - 0.357T + 61T^{2} \)
67 \( 1 - 2.78T + 67T^{2} \)
71 \( 1 + 4.26T + 71T^{2} \)
73 \( 1 + 4.79T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.57112164342361237183377167183, −6.63201890466099757520272311699, −6.02007430005121523276038299054, −5.45182392625491508121110887934, −4.92902979393269576619077629554, −3.61506515233147660857320692594, −3.40243938272232509883891007197, −2.05767977182238800906131635308, −1.28609591188360719363498497339, 0, 1.28609591188360719363498497339, 2.05767977182238800906131635308, 3.40243938272232509883891007197, 3.61506515233147660857320692594, 4.92902979393269576619077629554, 5.45182392625491508121110887934, 6.02007430005121523276038299054, 6.63201890466099757520272311699, 7.57112164342361237183377167183

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