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  + 2.15·3-s + 1.53·5-s − 0.745·7-s + 1.62·9-s − 4.48·11-s + 2.05·13-s + 3.30·15-s − 17-s − 5.76·19-s − 1.60·21-s − 4.38·23-s − 2.63·25-s − 2.95·27-s + 8.97·29-s − 0.172·31-s − 9.64·33-s − 1.14·35-s + 6.69·37-s + 4.41·39-s + 1.53·41-s − 3.47·43-s + 2.49·45-s − 10.8·47-s − 6.44·49-s − 2.15·51-s + 3.84·53-s − 6.90·55-s + ⋯
L(s)  = 1  + 1.24·3-s + 0.688·5-s − 0.281·7-s + 0.541·9-s − 1.35·11-s + 0.569·13-s + 0.854·15-s − 0.242·17-s − 1.32·19-s − 0.349·21-s − 0.914·23-s − 0.526·25-s − 0.569·27-s + 1.66·29-s − 0.0309·31-s − 1.67·33-s − 0.193·35-s + 1.10·37-s + 0.706·39-s + 0.239·41-s − 0.529·43-s + 0.372·45-s − 1.57·47-s − 0.920·49-s − 0.301·51-s + 0.528·53-s − 0.930·55-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 \( 1 - 2.15T + 3T^{2} \)
5 \( 1 - 1.53T + 5T^{2} \)
7 \( 1 + 0.745T + 7T^{2} \)
11 \( 1 + 4.48T + 11T^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
19 \( 1 + 5.76T + 19T^{2} \)
23 \( 1 + 4.38T + 23T^{2} \)
29 \( 1 - 8.97T + 29T^{2} \)
31 \( 1 + 0.172T + 31T^{2} \)
37 \( 1 - 6.69T + 37T^{2} \)
41 \( 1 - 1.53T + 41T^{2} \)
43 \( 1 + 3.47T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 3.84T + 53T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + 3.83T + 71T^{2} \)
73 \( 1 + 7.18T + 73T^{2} \)
79 \( 1 + 9.93T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−7.84624700820526265269789872630, −6.72748298746184036532429612137, −6.19014445739452968249731845734, −5.45787572797351813391755789438, −4.52717911907037210812964702375, −3.79972872103330323188742106399, −2.83240342867198430044281506404, −2.44152745290629393065288378378, −1.61544877776204764142287168069, 0, 1.61544877776204764142287168069, 2.44152745290629393065288378378, 2.83240342867198430044281506404, 3.79972872103330323188742106399, 4.52717911907037210812964702375, 5.45787572797351813391755789438, 6.19014445739452968249731845734, 6.72748298746184036532429612137, 7.84624700820526265269789872630

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