Properties

Degree 2
Conductor $ 2^{3} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.49·3-s + 1.95·5-s − 4.15·7-s + 3.22·9-s + 5.07·11-s − 3.39·13-s + 4.88·15-s − 17-s − 7.96·19-s − 10.3·21-s − 1.52·23-s − 1.16·25-s + 0.559·27-s + 2.94·29-s − 9.57·31-s + 12.6·33-s − 8.14·35-s − 3.43·37-s − 8.47·39-s + 1.52·41-s − 0.867·43-s + 6.31·45-s − 1.96·47-s + 10.2·49-s − 2.49·51-s + 3.60·53-s + 9.93·55-s + ⋯
L(s)  = 1  + 1.44·3-s + 0.875·5-s − 1.57·7-s + 1.07·9-s + 1.52·11-s − 0.941·13-s + 1.26·15-s − 0.242·17-s − 1.82·19-s − 2.26·21-s − 0.317·23-s − 0.232·25-s + 0.107·27-s + 0.546·29-s − 1.71·31-s + 2.20·33-s − 1.37·35-s − 0.564·37-s − 1.35·39-s + 0.237·41-s − 0.132·43-s + 0.941·45-s − 0.286·47-s + 1.46·49-s − 0.349·51-s + 0.495·53-s + 1.34·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8024,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 + T \)
good3 \( 1 - 2.49T + 3T^{2} \)
5 \( 1 - 1.95T + 5T^{2} \)
7 \( 1 + 4.15T + 7T^{2} \)
11 \( 1 - 5.07T + 11T^{2} \)
13 \( 1 + 3.39T + 13T^{2} \)
19 \( 1 + 7.96T + 19T^{2} \)
23 \( 1 + 1.52T + 23T^{2} \)
29 \( 1 - 2.94T + 29T^{2} \)
31 \( 1 + 9.57T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 - 1.52T + 41T^{2} \)
43 \( 1 + 0.867T + 43T^{2} \)
47 \( 1 + 1.96T + 47T^{2} \)
53 \( 1 - 3.60T + 53T^{2} \)
61 \( 1 + 2.74T + 61T^{2} \)
67 \( 1 - 16.0T + 67T^{2} \)
71 \( 1 - 1.96T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 9.09T + 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 - 2.26T + 89T^{2} \)
97 \( 1 - 4.97T + 97T^{2} \)
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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

−7.37051925272639338911449418991, −6.80121170536252211512578290752, −6.30205657043062297788042467007, −5.57866774373417194761134862544, −4.26442542166365785529802361631, −3.82514766585274922277790308045, −3.02636157373010186430816521907, −2.27779372600050600744403829115, −1.70316375004063620955654181229, 0, 1.70316375004063620955654181229, 2.27779372600050600744403829115, 3.02636157373010186430816521907, 3.82514766585274922277790308045, 4.26442542166365785529802361631, 5.57866774373417194761134862544, 6.30205657043062297788042467007, 6.80121170536252211512578290752, 7.37051925272639338911449418991

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