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  + 2.80·3-s − 3.33·5-s + 2.85·7-s + 4.87·9-s + 6.16·11-s − 3.18·13-s − 9.34·15-s − 3.44·17-s − 6.59·19-s + 8.01·21-s − 9.22·23-s + 6.09·25-s + 5.27·27-s + 1.40·29-s − 0.702·31-s + 17.2·33-s − 9.51·35-s − 9.54·37-s − 8.93·39-s − 2.82·41-s − 8.28·43-s − 16.2·45-s − 10.2·47-s + 1.16·49-s − 9.67·51-s − 5.03·53-s − 20.5·55-s + ⋯
L(s)  = 1  + 1.62·3-s − 1.48·5-s + 1.07·7-s + 1.62·9-s + 1.85·11-s − 0.882·13-s − 2.41·15-s − 0.836·17-s − 1.51·19-s + 1.75·21-s − 1.92·23-s + 1.21·25-s + 1.01·27-s + 0.260·29-s − 0.126·31-s + 3.01·33-s − 1.60·35-s − 1.56·37-s − 1.43·39-s − 0.441·41-s − 1.26·43-s − 2.42·45-s − 1.49·47-s + 0.166·49-s − 1.35·51-s − 0.691·53-s − 2.76·55-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 \( 1 - 2.80T + 3T^{2} \)
5 \( 1 + 3.33T + 5T^{2} \)
7 \( 1 - 2.85T + 7T^{2} \)
11 \( 1 - 6.16T + 11T^{2} \)
13 \( 1 + 3.18T + 13T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
19 \( 1 + 6.59T + 19T^{2} \)
23 \( 1 + 9.22T + 23T^{2} \)
29 \( 1 - 1.40T + 29T^{2} \)
31 \( 1 + 0.702T + 31T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + 2.82T + 41T^{2} \)
43 \( 1 + 8.28T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 5.03T + 53T^{2} \)
59 \( 1 - 3.57T + 59T^{2} \)
61 \( 1 - 2.46T + 61T^{2} \)
67 \( 1 + 0.271T + 67T^{2} \)
71 \( 1 - 6.18T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 7.67T + 79T^{2} \)
83 \( 1 + 1.85T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 + 0.763T + 97T^{2} \)
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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

−7.83905960758360221622316743428, −6.82709151253750848973903082323, −6.62146158914302325158313126897, −4.99373789614642235444073129468, −4.23804951798014758262408057551, −3.97685722065148951268309920161, −3.30457637641070027422658644464, −2.07017138769883736619352943805, −1.70624854843769854778435334651, 0, 1.70624854843769854778435334651, 2.07017138769883736619352943805, 3.30457637641070027422658644464, 3.97685722065148951268309920161, 4.23804951798014758262408057551, 4.99373789614642235444073129468, 6.62146158914302325158313126897, 6.82709151253750848973903082323, 7.83905960758360221622316743428

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