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  − 0.682·3-s + 3.69·5-s + 3.46·7-s − 2.53·9-s − 1.14·11-s − 0.915·13-s − 2.52·15-s − 7.09·17-s + 3.54·19-s − 2.36·21-s − 4.97·23-s + 8.65·25-s + 3.77·27-s − 4.41·29-s − 10.1·31-s + 0.781·33-s + 12.8·35-s + 2.60·37-s + 0.624·39-s − 9.41·41-s − 0.391·43-s − 9.36·45-s + 4.45·47-s + 5.01·49-s + 4.84·51-s − 1.57·53-s − 4.23·55-s + ⋯
L(s)  = 1  − 0.393·3-s + 1.65·5-s + 1.31·7-s − 0.844·9-s − 0.345·11-s − 0.254·13-s − 0.651·15-s − 1.72·17-s + 0.812·19-s − 0.516·21-s − 1.03·23-s + 1.73·25-s + 0.726·27-s − 0.820·29-s − 1.82·31-s + 0.136·33-s + 2.16·35-s + 0.428·37-s + 0.100·39-s − 1.47·41-s − 0.0597·43-s − 1.39·45-s + 0.650·47-s + 0.716·49-s + 0.678·51-s − 0.216·53-s − 0.570·55-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 \( 1 + 0.682T + 3T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + 1.14T + 11T^{2} \)
13 \( 1 + 0.915T + 13T^{2} \)
17 \( 1 + 7.09T + 17T^{2} \)
19 \( 1 - 3.54T + 19T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + 4.41T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 2.60T + 37T^{2} \)
41 \( 1 + 9.41T + 41T^{2} \)
43 \( 1 + 0.391T + 43T^{2} \)
47 \( 1 - 4.45T + 47T^{2} \)
53 \( 1 + 1.57T + 53T^{2} \)
59 \( 1 - 3.21T + 59T^{2} \)
61 \( 1 + 5.61T + 61T^{2} \)
67 \( 1 - 0.0216T + 67T^{2} \)
71 \( 1 - 1.49T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 - 0.0862T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−7.45264625743824337092539057774, −6.65984526807490432177143609124, −5.93251833960092697747366735462, −5.36536036270382640424699920350, −5.02931200821372670632842920327, −4.06741482574898590969798930259, −2.80261700259308324972320402381, −2.05437066907073049155435535811, −1.56947505273624099826573715630, 0, 1.56947505273624099826573715630, 2.05437066907073049155435535811, 2.80261700259308324972320402381, 4.06741482574898590969798930259, 5.02931200821372670632842920327, 5.36536036270382640424699920350, 5.93251833960092697747366735462, 6.65984526807490432177143609124, 7.45264625743824337092539057774

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