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  − 0.315·3-s + 2.25·5-s + 3.20·7-s − 2.90·9-s + 0.218·11-s − 4.17·13-s − 0.710·15-s − 4.68·17-s − 3.43·19-s − 1.01·21-s + 3.99·23-s + 0.0635·25-s + 1.86·27-s − 0.712·29-s − 1.04·31-s − 0.0688·33-s + 7.21·35-s + 1.70·37-s + 1.31·39-s + 3.18·41-s + 6.35·43-s − 6.52·45-s + 3.87·47-s + 3.28·49-s + 1.48·51-s − 10.8·53-s + 0.490·55-s + ⋯
L(s)  = 1  − 0.182·3-s + 1.00·5-s + 1.21·7-s − 0.966·9-s + 0.0657·11-s − 1.15·13-s − 0.183·15-s − 1.13·17-s − 0.789·19-s − 0.220·21-s + 0.833·23-s + 0.0127·25-s + 0.358·27-s − 0.132·29-s − 0.188·31-s − 0.0119·33-s + 1.21·35-s + 0.281·37-s + 0.211·39-s + 0.497·41-s + 0.969·43-s − 0.972·45-s + 0.565·47-s + 0.468·49-s + 0.207·51-s − 1.49·53-s + 0.0661·55-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 \( 1 + 0.315T + 3T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 - 0.218T + 11T^{2} \)
13 \( 1 + 4.17T + 13T^{2} \)
17 \( 1 + 4.68T + 17T^{2} \)
19 \( 1 + 3.43T + 19T^{2} \)
23 \( 1 - 3.99T + 23T^{2} \)
29 \( 1 + 0.712T + 29T^{2} \)
31 \( 1 + 1.04T + 31T^{2} \)
37 \( 1 - 1.70T + 37T^{2} \)
41 \( 1 - 3.18T + 41T^{2} \)
43 \( 1 - 6.35T + 43T^{2} \)
47 \( 1 - 3.87T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 2.63T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 6.09T + 67T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 + 2.78T + 73T^{2} \)
79 \( 1 + 5.16T + 79T^{2} \)
83 \( 1 + 1.83T + 83T^{2} \)
89 \( 1 + 8.08T + 89T^{2} \)
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

−7.47929626886847559631619591270, −6.74578357969971850280992101775, −5.98885938078956573333093326862, −5.40069414922810508461805632154, −4.78479852578347896468035227216, −4.14508453765839415100035774354, −2.71136164798044342617013109858, −2.31255365548548628294672534861, −1.41546261645210980502326801104, 0, 1.41546261645210980502326801104, 2.31255365548548628294672534861, 2.71136164798044342617013109858, 4.14508453765839415100035774354, 4.78479852578347896468035227216, 5.40069414922810508461805632154, 5.98885938078956573333093326862, 6.74578357969971850280992101775, 7.47929626886847559631619591270

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