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  − 2.78·2-s − 2.83·3-s + 5.72·4-s − 1.35·5-s + 7.88·6-s + 4.56·7-s − 10.3·8-s + 5.03·9-s + 3.77·10-s − 2.32·11-s − 16.2·12-s + 0.925·13-s − 12.6·14-s + 3.84·15-s + 17.3·16-s + 6.55·17-s − 14.0·18-s + 3.74·19-s − 7.78·20-s − 12.9·21-s + 6.45·22-s − 3.65·23-s + 29.3·24-s − 3.15·25-s − 2.57·26-s − 5.77·27-s + 26.1·28-s + ⋯
L(s)  = 1  − 1.96·2-s − 1.63·3-s + 2.86·4-s − 0.607·5-s + 3.21·6-s + 1.72·7-s − 3.66·8-s + 1.67·9-s + 1.19·10-s − 0.699·11-s − 4.68·12-s + 0.256·13-s − 3.39·14-s + 0.994·15-s + 4.34·16-s + 1.58·17-s − 3.30·18-s + 0.858·19-s − 1.73·20-s − 2.82·21-s + 1.37·22-s − 0.761·23-s + 6.00·24-s − 0.631·25-s − 0.504·26-s − 1.11·27-s + 4.94·28-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 \( 1 + 2.78T + 2T^{2} \)
3 \( 1 + 2.83T + 3T^{2} \)
5 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 - 4.56T + 7T^{2} \)
11 \( 1 + 2.32T + 11T^{2} \)
13 \( 1 - 0.925T + 13T^{2} \)
17 \( 1 - 6.55T + 17T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 6.80T + 29T^{2} \)
31 \( 1 - 0.567T + 31T^{2} \)
37 \( 1 + 0.527T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 - 6.93T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 - 0.248T + 67T^{2} \)
73 \( 1 + 4.69T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.48579447026840996690932145341, −7.23546638291304038270083856167, −6.21917098225623820859591933773, −5.52153372464739569888394698039, −5.11646622786527353992210942892, −3.90972823547577038829686158504, −2.67099975981658973795231291266, −1.47517326782295072814045554824, −1.06368369263331774634342495413, 0, 1.06368369263331774634342495413, 1.47517326782295072814045554824, 2.67099975981658973795231291266, 3.90972823547577038829686158504, 5.11646622786527353992210942892, 5.52153372464739569888394698039, 6.21917098225623820859591933773, 7.23546638291304038270083856167, 7.48579447026840996690932145341

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