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  − 3.11·3-s + 4.12·5-s + 1.17·7-s + 6.69·9-s − 4.55·11-s − 2.99·13-s − 12.8·15-s − 17-s − 3.45·19-s − 3.65·21-s − 0.461·23-s + 12.0·25-s − 11.5·27-s + 5.51·29-s + 0.588·31-s + 14.1·33-s + 4.83·35-s + 3.02·37-s + 9.32·39-s − 4.05·41-s + 8.04·43-s + 27.6·45-s + 1.26·47-s − 5.62·49-s + 3.11·51-s − 5.96·53-s − 18.7·55-s + ⋯
L(s)  = 1  − 1.79·3-s + 1.84·5-s + 0.443·7-s + 2.23·9-s − 1.37·11-s − 0.830·13-s − 3.31·15-s − 0.242·17-s − 0.793·19-s − 0.796·21-s − 0.0961·23-s + 2.40·25-s − 2.21·27-s + 1.02·29-s + 0.105·31-s + 2.46·33-s + 0.817·35-s + 0.496·37-s + 1.49·39-s − 0.632·41-s + 1.22·43-s + 4.12·45-s + 0.184·47-s − 0.803·49-s + 0.436·51-s − 0.819·53-s − 2.53·55-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 \( 1 + 3.11T + 3T^{2} \)
5 \( 1 - 4.12T + 5T^{2} \)
7 \( 1 - 1.17T + 7T^{2} \)
11 \( 1 + 4.55T + 11T^{2} \)
13 \( 1 + 2.99T + 13T^{2} \)
19 \( 1 + 3.45T + 19T^{2} \)
23 \( 1 + 0.461T + 23T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 - 0.588T + 31T^{2} \)
37 \( 1 - 3.02T + 37T^{2} \)
41 \( 1 + 4.05T + 41T^{2} \)
43 \( 1 - 8.04T + 43T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 + 5.96T + 53T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 + 9.06T + 67T^{2} \)
71 \( 1 + 0.0811T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 0.604T + 79T^{2} \)
83 \( 1 + 1.82T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 2.19T + 97T^{2} \)
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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

−7.19818723321063880428284664387, −6.54473165994262621059132680183, −5.96940813596273431951636757588, −5.49258392369651740222919392336, −4.85821770691718262856272601870, −4.51088643362973817940411045704, −2.76116988978392955487799466580, −2.08485233634353434718592044280, −1.19452328392631576136237346938, 0, 1.19452328392631576136237346938, 2.08485233634353434718592044280, 2.76116988978392955487799466580, 4.51088643362973817940411045704, 4.85821770691718262856272601870, 5.49258392369651740222919392336, 5.96940813596273431951636757588, 6.54473165994262621059132680183, 7.19818723321063880428284664387

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