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  − 2.64·3-s + 3.09·5-s − 1.88·7-s + 3.97·9-s + 0.597·11-s + 2.98·13-s − 8.18·15-s − 17-s + 4.89·19-s + 4.96·21-s − 4.83·23-s + 4.60·25-s − 2.56·27-s − 2.75·29-s − 5.24·31-s − 1.57·33-s − 5.83·35-s + 1.62·37-s − 7.87·39-s + 0.567·41-s − 4.98·43-s + 12.3·45-s − 13.6·47-s − 3.45·49-s + 2.64·51-s + 12.5·53-s + 1.85·55-s + ⋯
L(s)  = 1  − 1.52·3-s + 1.38·5-s − 0.711·7-s + 1.32·9-s + 0.180·11-s + 0.827·13-s − 2.11·15-s − 0.242·17-s + 1.12·19-s + 1.08·21-s − 1.00·23-s + 0.920·25-s − 0.492·27-s − 0.511·29-s − 0.942·31-s − 0.274·33-s − 0.985·35-s + 0.267·37-s − 1.26·39-s + 0.0885·41-s − 0.759·43-s + 1.83·45-s − 1.98·47-s − 0.494·49-s + 0.369·51-s + 1.72·53-s + 0.249·55-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 \( 1 + 2.64T + 3T^{2} \)
5 \( 1 - 3.09T + 5T^{2} \)
7 \( 1 + 1.88T + 7T^{2} \)
11 \( 1 - 0.597T + 11T^{2} \)
13 \( 1 - 2.98T + 13T^{2} \)
19 \( 1 - 4.89T + 19T^{2} \)
23 \( 1 + 4.83T + 23T^{2} \)
29 \( 1 + 2.75T + 29T^{2} \)
31 \( 1 + 5.24T + 31T^{2} \)
37 \( 1 - 1.62T + 37T^{2} \)
41 \( 1 - 0.567T + 41T^{2} \)
43 \( 1 + 4.98T + 43T^{2} \)
47 \( 1 + 13.6T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
61 \( 1 + 3.62T + 61T^{2} \)
67 \( 1 - 3.19T + 67T^{2} \)
71 \( 1 + 0.921T + 71T^{2} \)
73 \( 1 + 6.23T + 73T^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 - 1.29T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−7.12375757439285199592346562118, −6.51602166871898276042018666278, −6.05719680739430653204659828916, −5.55849140647726133597710262031, −5.03609758234616574144863625298, −3.99712062682214909267280276917, −3.13970582834571286313667619086, −1.94557337635968248488608538771, −1.20490125433407660414199199464, 0, 1.20490125433407660414199199464, 1.94557337635968248488608538771, 3.13970582834571286313667619086, 3.99712062682214909267280276917, 5.03609758234616574144863625298, 5.55849140647726133597710262031, 6.05719680739430653204659828916, 6.51602166871898276042018666278, 7.12375757439285199592346562118

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