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  + 3.07·3-s + 0.386·5-s − 0.194·7-s + 6.43·9-s − 2.36·11-s − 1.22·13-s + 1.18·15-s − 5.04·17-s − 4.24·19-s − 0.598·21-s − 1.53·23-s − 4.85·25-s + 10.5·27-s − 7.31·29-s − 3.33·31-s − 7.26·33-s − 0.0752·35-s − 2.17·37-s − 3.76·39-s − 1.04·41-s + 1.53·43-s + 2.48·45-s + 1.08·47-s − 6.96·49-s − 15.5·51-s + 1.55·53-s − 0.912·55-s + ⋯
L(s)  = 1  + 1.77·3-s + 0.172·5-s − 0.0736·7-s + 2.14·9-s − 0.712·11-s − 0.340·13-s + 0.306·15-s − 1.22·17-s − 0.973·19-s − 0.130·21-s − 0.319·23-s − 0.970·25-s + 2.03·27-s − 1.35·29-s − 0.599·31-s − 1.26·33-s − 0.0127·35-s − 0.356·37-s − 0.603·39-s − 0.163·41-s + 0.233·43-s + 0.370·45-s + 0.158·47-s − 0.994·49-s − 2.17·51-s + 0.213·53-s − 0.123·55-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 \( 1 - 3.07T + 3T^{2} \)
5 \( 1 - 0.386T + 5T^{2} \)
7 \( 1 + 0.194T + 7T^{2} \)
11 \( 1 + 2.36T + 11T^{2} \)
13 \( 1 + 1.22T + 13T^{2} \)
17 \( 1 + 5.04T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 1.53T + 23T^{2} \)
29 \( 1 + 7.31T + 29T^{2} \)
31 \( 1 + 3.33T + 31T^{2} \)
37 \( 1 + 2.17T + 37T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 - 1.53T + 43T^{2} \)
47 \( 1 - 1.08T + 47T^{2} \)
53 \( 1 - 1.55T + 53T^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 + 5.45T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 0.902T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 6.10T + 83T^{2} \)
89 \( 1 - 6.44T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.73848767959801849284298789791, −6.99479141136372904470870036725, −6.27890971009660680523379852497, −5.27550471745736313642176598754, −4.39074272623757371323332052234, −3.80995116944800720540927464681, −3.02665180558042759351881926450, −2.12452960313394461885542522535, −1.89193302942981337190482801975, 0, 1.89193302942981337190482801975, 2.12452960313394461885542522535, 3.02665180558042759351881926450, 3.80995116944800720540927464681, 4.39074272623757371323332052234, 5.27550471745736313642176598754, 6.27890971009660680523379852497, 6.99479141136372904470870036725, 7.73848767959801849284298789791

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