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  + 0.0730·3-s − 4.10·5-s + 1.52·7-s − 2.99·9-s − 0.141·11-s + 5.15·13-s − 0.300·15-s − 0.728·17-s − 4.77·19-s + 0.111·21-s + 3.29·23-s + 11.8·25-s − 0.438·27-s + 5.05·29-s − 6.69·31-s − 0.0103·33-s − 6.25·35-s + 0.889·37-s + 0.376·39-s − 3.88·41-s − 10.3·43-s + 12.3·45-s + 10.1·47-s − 4.68·49-s − 0.0532·51-s − 5.37·53-s + 0.580·55-s + ⋯
L(s)  = 1  + 0.0421·3-s − 1.83·5-s + 0.575·7-s − 0.998·9-s − 0.0426·11-s + 1.43·13-s − 0.0775·15-s − 0.176·17-s − 1.09·19-s + 0.0242·21-s + 0.687·23-s + 2.37·25-s − 0.0842·27-s + 0.938·29-s − 1.20·31-s − 0.00179·33-s − 1.05·35-s + 0.146·37-s + 0.0603·39-s − 0.607·41-s − 1.57·43-s + 1.83·45-s + 1.47·47-s − 0.669·49-s − 0.00745·51-s − 0.738·53-s + 0.0783·55-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 \( 1 - 0.0730T + 3T^{2} \)
5 \( 1 + 4.10T + 5T^{2} \)
7 \( 1 - 1.52T + 7T^{2} \)
11 \( 1 + 0.141T + 11T^{2} \)
13 \( 1 - 5.15T + 13T^{2} \)
17 \( 1 + 0.728T + 17T^{2} \)
19 \( 1 + 4.77T + 19T^{2} \)
23 \( 1 - 3.29T + 23T^{2} \)
29 \( 1 - 5.05T + 29T^{2} \)
31 \( 1 + 6.69T + 31T^{2} \)
37 \( 1 - 0.889T + 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 5.37T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 2.06T + 61T^{2} \)
67 \( 1 + 0.840T + 67T^{2} \)
71 \( 1 - 3.16T + 71T^{2} \)
73 \( 1 - 1.65T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 - 4.48T + 83T^{2} \)
89 \( 1 - 9.40T + 89T^{2} \)
97 \( 1 - 3.59T + 97T^{2} \)
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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

−7.67763546045094113005231834883, −6.81025553881888899858358180795, −6.25890482253355710549033583792, −5.20378709586255679575928695505, −4.63783224006608136903353296686, −3.68749627163571482422255689521, −3.45053498185534774114993857415, −2.32092539404316576061158008418, −1.04159619961036427729963435093, 0, 1.04159619961036427729963435093, 2.32092539404316576061158008418, 3.45053498185534774114993857415, 3.68749627163571482422255689521, 4.63783224006608136903353296686, 5.20378709586255679575928695505, 6.25890482253355710549033583792, 6.81025553881888899858358180795, 7.67763546045094113005231834883

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