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  − 0.0763·3-s + 1.17·5-s − 0.469·7-s − 2.99·9-s + 5.74·11-s − 1.85·13-s − 0.0895·15-s + 5.22·17-s − 2.12·19-s + 0.0358·21-s − 0.171·23-s − 3.62·25-s + 0.457·27-s − 6.19·29-s + 0.396·31-s − 0.438·33-s − 0.550·35-s − 8.17·37-s + 0.141·39-s − 12.4·41-s + 4.97·43-s − 3.51·45-s + 0.521·47-s − 6.77·49-s − 0.399·51-s + 8.76·53-s + 6.73·55-s + ⋯
L(s)  = 1  − 0.0440·3-s + 0.524·5-s − 0.177·7-s − 0.998·9-s + 1.73·11-s − 0.515·13-s − 0.0231·15-s + 1.26·17-s − 0.487·19-s + 0.00782·21-s − 0.0358·23-s − 0.724·25-s + 0.0880·27-s − 1.14·29-s + 0.0711·31-s − 0.0763·33-s − 0.0930·35-s − 1.34·37-s + 0.0227·39-s − 1.94·41-s + 0.759·43-s − 0.523·45-s + 0.0760·47-s − 0.968·49-s − 0.0559·51-s + 1.20·53-s + 0.908·55-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 \( 1 + 0.0763T + 3T^{2} \)
5 \( 1 - 1.17T + 5T^{2} \)
7 \( 1 + 0.469T + 7T^{2} \)
11 \( 1 - 5.74T + 11T^{2} \)
13 \( 1 + 1.85T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 + 2.12T + 19T^{2} \)
23 \( 1 + 0.171T + 23T^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
31 \( 1 - 0.396T + 31T^{2} \)
37 \( 1 + 8.17T + 37T^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 - 4.97T + 43T^{2} \)
47 \( 1 - 0.521T + 47T^{2} \)
53 \( 1 - 8.76T + 53T^{2} \)
59 \( 1 + 3.35T + 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 + 8.42T + 67T^{2} \)
71 \( 1 + 7.47T + 71T^{2} \)
73 \( 1 + 4.60T + 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + 5.97T + 83T^{2} \)
89 \( 1 + 4.25T + 89T^{2} \)
97 \( 1 + 2.64T + 97T^{2} \)
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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

−7.41316063409238546946471360434, −6.73091075530571326467004479418, −6.00661499864123596420927921796, −5.59476336404322237565233754429, −4.72931879358692407899781319625, −3.70915906663825062570534642815, −3.26142099972572349957497134530, −2.11173797827183793348824636030, −1.37955790277592905801675394156, 0, 1.37955790277592905801675394156, 2.11173797827183793348824636030, 3.26142099972572349957497134530, 3.70915906663825062570534642815, 4.72931879358692407899781319625, 5.59476336404322237565233754429, 6.00661499864123596420927921796, 6.73091075530571326467004479418, 7.41316063409238546946471360434

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