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  − 3.24·3-s − 3.27·5-s + 3.90·7-s + 7.55·9-s + 3.39·11-s + 0.485·13-s + 10.6·15-s − 17-s − 3.70·19-s − 12.6·21-s + 6.21·23-s + 5.71·25-s − 14.8·27-s + 3.88·29-s − 4.41·31-s − 11.0·33-s − 12.7·35-s − 10.0·37-s − 1.57·39-s − 1.01·41-s + 10.8·43-s − 24.7·45-s − 11.2·47-s + 8.25·49-s + 3.24·51-s − 9.21·53-s − 11.1·55-s + ⋯
L(s)  = 1  − 1.87·3-s − 1.46·5-s + 1.47·7-s + 2.51·9-s + 1.02·11-s + 0.134·13-s + 2.74·15-s − 0.242·17-s − 0.849·19-s − 2.76·21-s + 1.29·23-s + 1.14·25-s − 2.85·27-s + 0.720·29-s − 0.793·31-s − 1.92·33-s − 2.16·35-s − 1.65·37-s − 0.252·39-s − 0.159·41-s + 1.64·43-s − 3.68·45-s − 1.64·47-s + 1.17·49-s + 0.455·51-s − 1.26·53-s − 1.50·55-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 \( 1 + 3.24T + 3T^{2} \)
5 \( 1 + 3.27T + 5T^{2} \)
7 \( 1 - 3.90T + 7T^{2} \)
11 \( 1 - 3.39T + 11T^{2} \)
13 \( 1 - 0.485T + 13T^{2} \)
19 \( 1 + 3.70T + 19T^{2} \)
23 \( 1 - 6.21T + 23T^{2} \)
29 \( 1 - 3.88T + 29T^{2} \)
31 \( 1 + 4.41T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 1.01T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + 9.21T + 53T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 0.0650T + 67T^{2} \)
71 \( 1 - 6.83T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 0.493T + 83T^{2} \)
89 \( 1 - 4.81T + 89T^{2} \)
97 \( 1 - 4.67T + 97T^{2} \)
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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

−7.27871752603291666921552509158, −6.83787189673250554630523282203, −6.14568495471977316870668205158, −5.19756200551824710947723244342, −4.69096671124403209351843723376, −4.27221515500802921261009552561, −3.47443482304946433918078407726, −1.75173586141286990161274596569, −1.04307449331364630874134043064, 0, 1.04307449331364630874134043064, 1.75173586141286990161274596569, 3.47443482304946433918078407726, 4.27221515500802921261009552561, 4.69096671124403209351843723376, 5.19756200551824710947723244342, 6.14568495471977316870668205158, 6.83787189673250554630523282203, 7.27871752603291666921552509158

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