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  + 1.73·3-s + 2.58·5-s − 2.10·7-s + 0.0181·9-s + 2.62·11-s + 0.715·13-s + 4.49·15-s − 5.38·17-s − 1.38·19-s − 3.65·21-s − 6.72·23-s + 1.68·25-s − 5.18·27-s − 4.19·29-s + 5.92·31-s + 4.55·33-s − 5.43·35-s − 7.49·37-s + 1.24·39-s − 2.29·41-s − 7.41·43-s + 0.0468·45-s + 7.28·47-s − 2.57·49-s − 9.34·51-s − 11.8·53-s + 6.78·55-s + ⋯
L(s)  = 1  + 1.00·3-s + 1.15·5-s − 0.794·7-s + 0.00604·9-s + 0.790·11-s + 0.198·13-s + 1.16·15-s − 1.30·17-s − 0.317·19-s − 0.797·21-s − 1.40·23-s + 0.337·25-s − 0.996·27-s − 0.778·29-s + 1.06·31-s + 0.793·33-s − 0.919·35-s − 1.23·37-s + 0.198·39-s − 0.357·41-s − 1.13·43-s + 0.00699·45-s + 1.06·47-s − 0.368·49-s − 1.30·51-s − 1.63·53-s + 0.914·55-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 \( 1 - 1.73T + 3T^{2} \)
5 \( 1 - 2.58T + 5T^{2} \)
7 \( 1 + 2.10T + 7T^{2} \)
11 \( 1 - 2.62T + 11T^{2} \)
13 \( 1 - 0.715T + 13T^{2} \)
17 \( 1 + 5.38T + 17T^{2} \)
19 \( 1 + 1.38T + 19T^{2} \)
23 \( 1 + 6.72T + 23T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 + 2.29T + 41T^{2} \)
43 \( 1 + 7.41T + 43T^{2} \)
47 \( 1 - 7.28T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 - 7.39T + 59T^{2} \)
61 \( 1 + 1.47T + 61T^{2} \)
67 \( 1 - 4.45T + 67T^{2} \)
71 \( 1 + 8.17T + 71T^{2} \)
73 \( 1 + 9.67T + 73T^{2} \)
79 \( 1 + 9.04T + 79T^{2} \)
83 \( 1 + 4.25T + 83T^{2} \)
89 \( 1 - 9.05T + 89T^{2} \)
97 \( 1 - 8.08T + 97T^{2} \)
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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

−7.51808529524351862914898085363, −6.58307187528995790896515022397, −6.27477884929173377581970568153, −5.58079860761869769829069616930, −4.50973143149963545169717647985, −3.73903630337895332255994765454, −3.05258688878418259498146557073, −2.15214773542203296996723571427, −1.71057175729013531333331852644, 0, 1.71057175729013531333331852644, 2.15214773542203296996723571427, 3.05258688878418259498146557073, 3.73903630337895332255994765454, 4.50973143149963545169717647985, 5.58079860761869769829069616930, 6.27477884929173377581970568153, 6.58307187528995790896515022397, 7.51808529524351862914898085363

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