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  − 0.823·3-s + 3.63·5-s + 3.42·7-s − 2.32·9-s − 1.66·11-s + 1.71·13-s − 2.99·15-s − 17-s − 8.21·19-s − 2.81·21-s + 1.40·23-s + 8.24·25-s + 4.38·27-s − 9.95·29-s − 6.20·31-s + 1.37·33-s + 12.4·35-s − 1.91·37-s − 1.41·39-s + 4.14·41-s − 7.78·43-s − 8.45·45-s + 3.59·47-s + 4.71·49-s + 0.823·51-s − 9.21·53-s − 6.07·55-s + ⋯
L(s)  = 1  − 0.475·3-s + 1.62·5-s + 1.29·7-s − 0.773·9-s − 0.503·11-s + 0.475·13-s − 0.773·15-s − 0.242·17-s − 1.88·19-s − 0.615·21-s + 0.293·23-s + 1.64·25-s + 0.843·27-s − 1.84·29-s − 1.11·31-s + 0.239·33-s + 2.10·35-s − 0.314·37-s − 0.226·39-s + 0.647·41-s − 1.18·43-s − 1.25·45-s + 0.524·47-s + 0.673·49-s + 0.115·51-s − 1.26·53-s − 0.819·55-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 \( 1 + 0.823T + 3T^{2} \)
5 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + 1.66T + 11T^{2} \)
13 \( 1 - 1.71T + 13T^{2} \)
19 \( 1 + 8.21T + 19T^{2} \)
23 \( 1 - 1.40T + 23T^{2} \)
29 \( 1 + 9.95T + 29T^{2} \)
31 \( 1 + 6.20T + 31T^{2} \)
37 \( 1 + 1.91T + 37T^{2} \)
41 \( 1 - 4.14T + 41T^{2} \)
43 \( 1 + 7.78T + 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 + 9.21T + 53T^{2} \)
61 \( 1 + 9.69T + 61T^{2} \)
67 \( 1 + 7.29T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 4.25T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 7.87T + 83T^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−7.53590213531335528033996951764, −6.48053143678569752814727548551, −6.07412691159735867152243605382, −5.36843939615508367711621976820, −5.00495747092808943081499864316, −4.06753378238644252868546630260, −2.86176501536207599151624196885, −1.97043571403121052710827279475, −1.59554641514916828341744469039, 0, 1.59554641514916828341744469039, 1.97043571403121052710827279475, 2.86176501536207599151624196885, 4.06753378238644252868546630260, 5.00495747092808943081499864316, 5.36843939615508367711621976820, 6.07412691159735867152243605382, 6.48053143678569752814727548551, 7.53590213531335528033996951764

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