Properties

Degree 2
Conductor $ 71 \cdot 113 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.49·2-s − 3.32·3-s + 4.23·4-s − 4.15·5-s + 8.30·6-s − 2.74·7-s − 5.58·8-s + 8.06·9-s + 10.3·10-s + 5.53·11-s − 14.0·12-s + 4.58·13-s + 6.85·14-s + 13.8·15-s + 5.47·16-s + 0.508·17-s − 20.1·18-s − 0.582·19-s − 17.6·20-s + 9.13·21-s − 13.8·22-s − 1.53·23-s + 18.5·24-s + 12.2·25-s − 11.4·26-s − 16.8·27-s − 11.6·28-s + ⋯
L(s)  = 1  − 1.76·2-s − 1.92·3-s + 2.11·4-s − 1.85·5-s + 3.39·6-s − 1.03·7-s − 1.97·8-s + 2.68·9-s + 3.28·10-s + 1.66·11-s − 4.06·12-s + 1.27·13-s + 1.83·14-s + 3.57·15-s + 1.36·16-s + 0.123·17-s − 4.74·18-s − 0.133·19-s − 3.93·20-s + 1.99·21-s − 2.94·22-s − 0.320·23-s + 3.79·24-s + 2.45·25-s − 2.24·26-s − 3.24·27-s − 2.19·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8023\)    =    \(71 \cdot 113\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8023} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8023,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{71,\;113\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{71,\;113\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad71 \( 1 + T \)
113 \( 1 + T \)
good2 \( 1 + 2.49T + 2T^{2} \)
3 \( 1 + 3.32T + 3T^{2} \)
5 \( 1 + 4.15T + 5T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 - 5.53T + 11T^{2} \)
13 \( 1 - 4.58T + 13T^{2} \)
17 \( 1 - 0.508T + 17T^{2} \)
19 \( 1 + 0.582T + 19T^{2} \)
23 \( 1 + 1.53T + 23T^{2} \)
29 \( 1 - 2.85T + 29T^{2} \)
31 \( 1 - 6.73T + 31T^{2} \)
37 \( 1 - 3.94T + 37T^{2} \)
41 \( 1 + 0.699T + 41T^{2} \)
43 \( 1 - 9.44T + 43T^{2} \)
47 \( 1 - 8.69T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 - 7.59T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 2.24T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 4.50T + 97T^{2} \)
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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

−7.41914062923641818399473406189, −6.85696406410579616974732507683, −6.32456013226300380076046228994, −6.00576903911602323617941114528, −4.43644365658634335860434651096, −4.07391730453814493117017907750, −3.12638749662962546144995970510, −1.29968031077455588937423073919, −0.851406460308151085586252268775, 0, 0.851406460308151085586252268775, 1.29968031077455588937423073919, 3.12638749662962546144995970510, 4.07391730453814493117017907750, 4.43644365658634335860434651096, 6.00576903911602323617941114528, 6.32456013226300380076046228994, 6.85696406410579616974732507683, 7.41914062923641818399473406189

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