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  − 1.66·3-s − 3.44·5-s − 4.43·7-s − 0.230·9-s − 1.18·11-s − 3.49·13-s + 5.72·15-s − 2.17·17-s − 5.57·19-s + 7.37·21-s + 7.37·23-s + 6.85·25-s + 5.37·27-s − 4.21·29-s + 7.45·31-s + 1.97·33-s + 15.2·35-s − 5.18·37-s + 5.82·39-s − 1.01·41-s + 0.120·43-s + 0.792·45-s − 3.07·47-s + 12.6·49-s + 3.61·51-s − 2.80·53-s + 4.09·55-s + ⋯
L(s)  = 1  − 0.960·3-s − 1.53·5-s − 1.67·7-s − 0.0767·9-s − 0.358·11-s − 0.969·13-s + 1.47·15-s − 0.526·17-s − 1.27·19-s + 1.60·21-s + 1.53·23-s + 1.37·25-s + 1.03·27-s − 0.783·29-s + 1.33·31-s + 0.344·33-s + 2.57·35-s − 0.852·37-s + 0.931·39-s − 0.159·41-s + 0.0183·43-s + 0.118·45-s − 0.448·47-s + 1.80·49-s + 0.506·51-s − 0.385·53-s + 0.551·55-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 \( 1 + 1.66T + 3T^{2} \)
5 \( 1 + 3.44T + 5T^{2} \)
7 \( 1 + 4.43T + 7T^{2} \)
11 \( 1 + 1.18T + 11T^{2} \)
13 \( 1 + 3.49T + 13T^{2} \)
17 \( 1 + 2.17T + 17T^{2} \)
19 \( 1 + 5.57T + 19T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 + 4.21T + 29T^{2} \)
31 \( 1 - 7.45T + 31T^{2} \)
37 \( 1 + 5.18T + 37T^{2} \)
41 \( 1 + 1.01T + 41T^{2} \)
43 \( 1 - 0.120T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 + 2.80T + 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 4.55T + 67T^{2} \)
71 \( 1 - 0.710T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 - 6.35T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 - 9.71T + 89T^{2} \)
97 \( 1 - 7.58T + 97T^{2} \)
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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

−7.28128484050077051607780458813, −6.68786875439998055473041382010, −6.37348002488417893117458602419, −5.28740926188208359670702642319, −4.73413926166936622258748191757, −3.91142390530039976644906685615, −3.17687725804833473117995310593, −2.49003628405549794142777630449, −0.60147958183965052945995551813, 0, 0.60147958183965052945995551813, 2.49003628405549794142777630449, 3.17687725804833473117995310593, 3.91142390530039976644906685615, 4.73413926166936622258748191757, 5.28740926188208359670702642319, 6.37348002488417893117458602419, 6.68786875439998055473041382010, 7.28128484050077051607780458813

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