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  − 1.59·3-s + 2.75·5-s − 1.46·7-s − 0.462·9-s + 5.86·11-s − 6.69·13-s − 4.38·15-s − 6.07·17-s + 1.48·19-s + 2.33·21-s − 4.40·23-s + 2.58·25-s + 5.51·27-s + 8.17·29-s + 8.84·31-s − 9.35·33-s − 4.03·35-s − 7.89·37-s + 10.6·39-s + 5.35·41-s + 8.36·43-s − 1.27·45-s − 9.07·47-s − 4.85·49-s + 9.67·51-s + 12.3·53-s + 16.1·55-s + ⋯
L(s)  = 1  − 0.919·3-s + 1.23·5-s − 0.553·7-s − 0.154·9-s + 1.76·11-s − 1.85·13-s − 1.13·15-s − 1.47·17-s + 0.340·19-s + 0.508·21-s − 0.919·23-s + 0.517·25-s + 1.06·27-s + 1.51·29-s + 1.58·31-s − 1.62·33-s − 0.681·35-s − 1.29·37-s + 1.70·39-s + 0.836·41-s + 1.27·43-s − 0.189·45-s − 1.32·47-s − 0.693·49-s + 1.35·51-s + 1.69·53-s + 2.18·55-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 \( 1 + 1.59T + 3T^{2} \)
5 \( 1 - 2.75T + 5T^{2} \)
7 \( 1 + 1.46T + 7T^{2} \)
11 \( 1 - 5.86T + 11T^{2} \)
13 \( 1 + 6.69T + 13T^{2} \)
17 \( 1 + 6.07T + 17T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
23 \( 1 + 4.40T + 23T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 - 8.84T + 31T^{2} \)
37 \( 1 + 7.89T + 37T^{2} \)
41 \( 1 - 5.35T + 41T^{2} \)
43 \( 1 - 8.36T + 43T^{2} \)
47 \( 1 + 9.07T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 - 2.57T + 59T^{2} \)
61 \( 1 - 3.03T + 61T^{2} \)
67 \( 1 + 1.68T + 67T^{2} \)
71 \( 1 + 4.21T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 - 17.2T + 83T^{2} \)
89 \( 1 + 8.75T + 89T^{2} \)
97 \( 1 - 2.54T + 97T^{2} \)
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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

−7.03928868416339725170757491273, −6.63715191077240545801847505109, −6.20303359083087446286255888979, −5.53220937919851126438080434505, −4.71896682310990434438816768733, −4.19145158346584009038112992317, −2.84713228312377016688145660781, −2.26294848691749785915379774451, −1.18369158068410716079231707868, 0, 1.18369158068410716079231707868, 2.26294848691749785915379774451, 2.84713228312377016688145660781, 4.19145158346584009038112992317, 4.71896682310990434438816768733, 5.53220937919851126438080434505, 6.20303359083087446286255888979, 6.63715191077240545801847505109, 7.03928868416339725170757491273

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