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  + 0.453·3-s + 2.43·5-s + 1.57·7-s − 2.79·9-s − 1.43·11-s − 3.06·13-s + 1.10·15-s − 17-s + 3.99·19-s + 0.715·21-s − 6.82·23-s + 0.939·25-s − 2.62·27-s − 3.52·29-s + 7.83·31-s − 0.650·33-s + 3.84·35-s − 6.03·37-s − 1.38·39-s + 1.39·41-s + 7.98·43-s − 6.81·45-s − 8.67·47-s − 4.51·49-s − 0.453·51-s + 1.22·53-s − 3.49·55-s + ⋯
L(s)  = 1  + 0.261·3-s + 1.08·5-s + 0.596·7-s − 0.931·9-s − 0.432·11-s − 0.849·13-s + 0.285·15-s − 0.242·17-s + 0.917·19-s + 0.156·21-s − 1.42·23-s + 0.187·25-s − 0.505·27-s − 0.655·29-s + 1.40·31-s − 0.113·33-s + 0.649·35-s − 0.992·37-s − 0.222·39-s + 0.217·41-s + 1.21·43-s − 1.01·45-s − 1.26·47-s − 0.644·49-s − 0.0634·51-s + 0.168·53-s − 0.471·55-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 \( 1 - 0.453T + 3T^{2} \)
5 \( 1 - 2.43T + 5T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 + 1.43T + 11T^{2} \)
13 \( 1 + 3.06T + 13T^{2} \)
19 \( 1 - 3.99T + 19T^{2} \)
23 \( 1 + 6.82T + 23T^{2} \)
29 \( 1 + 3.52T + 29T^{2} \)
31 \( 1 - 7.83T + 31T^{2} \)
37 \( 1 + 6.03T + 37T^{2} \)
41 \( 1 - 1.39T + 41T^{2} \)
43 \( 1 - 7.98T + 43T^{2} \)
47 \( 1 + 8.67T + 47T^{2} \)
53 \( 1 - 1.22T + 53T^{2} \)
61 \( 1 - 5.68T + 61T^{2} \)
67 \( 1 + 5.05T + 67T^{2} \)
71 \( 1 - 3.31T + 71T^{2} \)
73 \( 1 - 3.07T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + 2.49T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−7.71646981954710134509323116674, −6.72775155859650486811084108718, −5.97337891393813811270883175097, −5.42365336699682071463907155933, −4.87651905990885825607187883604, −3.88303351750914221930778503460, −2.82477931265470061761084355316, −2.31108928444899979402357407075, −1.48772203313641968810649845750, 0, 1.48772203313641968810649845750, 2.31108928444899979402357407075, 2.82477931265470061761084355316, 3.88303351750914221930778503460, 4.87651905990885825607187883604, 5.42365336699682071463907155933, 5.97337891393813811270883175097, 6.72775155859650486811084108718, 7.71646981954710134509323116674

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