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  + 1.48·3-s − 1.79·5-s − 0.552·7-s − 0.782·9-s − 4.33·11-s + 2.54·13-s − 2.67·15-s + 2.52·17-s + 5.11·19-s − 0.822·21-s + 3.78·23-s − 1.76·25-s − 5.63·27-s + 0.907·29-s − 0.380·31-s − 6.45·33-s + 0.993·35-s − 5.43·37-s + 3.79·39-s + 5.72·41-s + 9.21·43-s + 1.40·45-s + 8.81·47-s − 6.69·49-s + 3.75·51-s − 6.46·53-s + 7.80·55-s + ⋯
L(s)  = 1  + 0.859·3-s − 0.804·5-s − 0.208·7-s − 0.260·9-s − 1.30·11-s + 0.706·13-s − 0.691·15-s + 0.611·17-s + 1.17·19-s − 0.179·21-s + 0.789·23-s − 0.352·25-s − 1.08·27-s + 0.168·29-s − 0.0683·31-s − 1.12·33-s + 0.167·35-s − 0.893·37-s + 0.607·39-s + 0.894·41-s + 1.40·43-s + 0.209·45-s + 1.28·47-s − 0.956·49-s + 0.526·51-s − 0.887·53-s + 1.05·55-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 \( 1 - 1.48T + 3T^{2} \)
5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 + 0.552T + 7T^{2} \)
11 \( 1 + 4.33T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 - 2.52T + 17T^{2} \)
19 \( 1 - 5.11T + 19T^{2} \)
23 \( 1 - 3.78T + 23T^{2} \)
29 \( 1 - 0.907T + 29T^{2} \)
31 \( 1 + 0.380T + 31T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 - 5.72T + 41T^{2} \)
43 \( 1 - 9.21T + 43T^{2} \)
47 \( 1 - 8.81T + 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + 3.40T + 59T^{2} \)
61 \( 1 + 1.06T + 61T^{2} \)
67 \( 1 - 0.253T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 + 7.76T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−7.71936554289249552639941023335, −7.13070509582877743462550404178, −5.93256741679144430000556069085, −5.48488471783372028158072767853, −4.55165901785664649826814966180, −3.66571082914906635313130345654, −3.09789936675195886587305886680, −2.55190422313249098132454860680, −1.25729578586778742184228759951, 0, 1.25729578586778742184228759951, 2.55190422313249098132454860680, 3.09789936675195886587305886680, 3.66571082914906635313130345654, 4.55165901785664649826814966180, 5.48488471783372028158072767853, 5.93256741679144430000556069085, 7.13070509582877743462550404178, 7.71936554289249552639941023335

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