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  − 1.95·3-s + 0.337·5-s − 0.830·7-s + 0.835·9-s + 5.82·11-s + 4.14·13-s − 0.660·15-s − 4.44·17-s − 1.68·19-s + 1.62·21-s + 6.61·23-s − 4.88·25-s + 4.23·27-s − 6.73·29-s − 0.271·31-s − 11.4·33-s − 0.280·35-s − 4.23·37-s − 8.12·39-s − 8.53·41-s + 0.824·43-s + 0.281·45-s − 8.45·47-s − 6.30·49-s + 8.70·51-s − 2.66·53-s + 1.96·55-s + ⋯
L(s)  = 1  − 1.13·3-s + 0.150·5-s − 0.313·7-s + 0.278·9-s + 1.75·11-s + 1.15·13-s − 0.170·15-s − 1.07·17-s − 0.387·19-s + 0.355·21-s + 1.38·23-s − 0.977·25-s + 0.815·27-s − 1.25·29-s − 0.0487·31-s − 1.98·33-s − 0.0473·35-s − 0.696·37-s − 1.30·39-s − 1.33·41-s + 0.125·43-s + 0.0419·45-s − 1.23·47-s − 0.901·49-s + 1.21·51-s − 0.365·53-s + 0.265·55-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 \( 1 + 1.95T + 3T^{2} \)
5 \( 1 - 0.337T + 5T^{2} \)
7 \( 1 + 0.830T + 7T^{2} \)
11 \( 1 - 5.82T + 11T^{2} \)
13 \( 1 - 4.14T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 + 1.68T + 19T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + 6.73T + 29T^{2} \)
31 \( 1 + 0.271T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + 8.53T + 41T^{2} \)
43 \( 1 - 0.824T + 43T^{2} \)
47 \( 1 + 8.45T + 47T^{2} \)
53 \( 1 + 2.66T + 53T^{2} \)
59 \( 1 - 5.99T + 59T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + 8.68T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 4.23T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−7.05443858475267100155818181159, −6.66474754860504983156802490671, −6.20810359891277084084018717315, −5.51654189534365290622851974849, −4.75989733678136319596973884554, −3.89594598138993985196785944962, −3.36461664780863573364165340169, −1.95642462309303095627666765915, −1.16508087988359572997944701976, 0, 1.16508087988359572997944701976, 1.95642462309303095627666765915, 3.36461664780863573364165340169, 3.89594598138993985196785944962, 4.75989733678136319596973884554, 5.51654189534365290622851974849, 6.20810359891277084084018717315, 6.66474754860504983156802490671, 7.05443858475267100155818181159

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