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  − 0.725·3-s − 2.56·5-s − 0.934·7-s − 2.47·9-s + 0.446·11-s + 5.91·13-s + 1.85·15-s − 17-s − 3.85·19-s + 0.677·21-s − 7.09·23-s + 1.56·25-s + 3.97·27-s + 9.38·29-s − 8.41·31-s − 0.324·33-s + 2.39·35-s + 5.65·37-s − 4.29·39-s + 8.92·41-s + 9.40·43-s + 6.33·45-s + 9.71·47-s − 6.12·49-s + 0.725·51-s − 4.68·53-s − 1.14·55-s + ⋯
L(s)  = 1  − 0.418·3-s − 1.14·5-s − 0.353·7-s − 0.824·9-s + 0.134·11-s + 1.64·13-s + 0.479·15-s − 0.242·17-s − 0.883·19-s + 0.147·21-s − 1.48·23-s + 0.312·25-s + 0.764·27-s + 1.74·29-s − 1.51·31-s − 0.0564·33-s + 0.404·35-s + 0.930·37-s − 0.687·39-s + 1.39·41-s + 1.43·43-s + 0.944·45-s + 1.41·47-s − 0.875·49-s + 0.101·51-s − 0.643·53-s − 0.154·55-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 \( 1 + 0.725T + 3T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 + 0.934T + 7T^{2} \)
11 \( 1 - 0.446T + 11T^{2} \)
13 \( 1 - 5.91T + 13T^{2} \)
19 \( 1 + 3.85T + 19T^{2} \)
23 \( 1 + 7.09T + 23T^{2} \)
29 \( 1 - 9.38T + 29T^{2} \)
31 \( 1 + 8.41T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 - 9.40T + 43T^{2} \)
47 \( 1 - 9.71T + 47T^{2} \)
53 \( 1 + 4.68T + 53T^{2} \)
61 \( 1 - 3.41T + 61T^{2} \)
67 \( 1 - 1.82T + 67T^{2} \)
71 \( 1 - 0.929T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 - 8.98T + 89T^{2} \)
97 \( 1 + 0.541T + 97T^{2} \)
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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

−7.64769713383309705699257305485, −6.62315461340670507783467768809, −6.09779328430011560310324564043, −5.64325958644212879256967546490, −4.34529437246675448215475588732, −4.06360036389659166739499778243, −3.24043309015810772240033575972, −2.32214002743137582059584930897, −0.969366534397503258965905729998, 0, 0.969366534397503258965905729998, 2.32214002743137582059584930897, 3.24043309015810772240033575972, 4.06360036389659166739499778243, 4.34529437246675448215475588732, 5.64325958644212879256967546490, 6.09779328430011560310324564043, 6.62315461340670507783467768809, 7.64769713383309705699257305485

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