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  − 1.74·3-s + 1.05·5-s + 1.84·7-s + 0.0425·9-s + 6.02·11-s − 0.598·13-s − 1.83·15-s − 17-s + 2.57·19-s − 3.21·21-s − 5.74·23-s − 3.88·25-s + 5.15·27-s − 2.33·29-s − 4.57·31-s − 10.5·33-s + 1.94·35-s + 8.60·37-s + 1.04·39-s − 10.8·41-s − 4.58·43-s + 0.0449·45-s + 2.41·47-s − 3.60·49-s + 1.74·51-s − 2.12·53-s + 6.35·55-s + ⋯
L(s)  = 1  − 1.00·3-s + 0.471·5-s + 0.696·7-s + 0.0141·9-s + 1.81·11-s − 0.166·13-s − 0.475·15-s − 0.242·17-s + 0.590·19-s − 0.701·21-s − 1.19·23-s − 0.777·25-s + 0.992·27-s − 0.433·29-s − 0.821·31-s − 1.82·33-s + 0.328·35-s + 1.41·37-s + 0.167·39-s − 1.68·41-s − 0.698·43-s + 0.00669·45-s + 0.352·47-s − 0.514·49-s + 0.244·51-s − 0.291·53-s + 0.856·55-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 \( 1 + 1.74T + 3T^{2} \)
5 \( 1 - 1.05T + 5T^{2} \)
7 \( 1 - 1.84T + 7T^{2} \)
11 \( 1 - 6.02T + 11T^{2} \)
13 \( 1 + 0.598T + 13T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 + 5.74T + 23T^{2} \)
29 \( 1 + 2.33T + 29T^{2} \)
31 \( 1 + 4.57T + 31T^{2} \)
37 \( 1 - 8.60T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 4.58T + 43T^{2} \)
47 \( 1 - 2.41T + 47T^{2} \)
53 \( 1 + 2.12T + 53T^{2} \)
61 \( 1 + 0.502T + 61T^{2} \)
67 \( 1 + 8.85T + 67T^{2} \)
71 \( 1 + 7.47T + 71T^{2} \)
73 \( 1 - 2.03T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + 2.18T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−7.30083592365129101792722945612, −6.68617136386519152818855663387, −5.90148492199974523950415182903, −5.68117987948398842397039787849, −4.66970801114370195996332147262, −4.13157301398854500204655496190, −3.17747300944606622834284709174, −1.88334977096772249294810658707, −1.33947776525169476071038213668, 0, 1.33947776525169476071038213668, 1.88334977096772249294810658707, 3.17747300944606622834284709174, 4.13157301398854500204655496190, 4.66970801114370195996332147262, 5.68117987948398842397039787849, 5.90148492199974523950415182903, 6.68617136386519152818855663387, 7.30083592365129101792722945612

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