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  − 2.73·2-s − 2.15·3-s + 5.46·4-s − 2.36·5-s + 5.88·6-s − 3.03·7-s − 9.45·8-s + 1.64·9-s + 6.46·10-s − 1.23·11-s − 11.7·12-s + 0.935·13-s + 8.30·14-s + 5.09·15-s + 14.8·16-s + 3.71·17-s − 4.48·18-s + 6.12·19-s − 12.9·20-s + 6.54·21-s + 3.37·22-s − 3.46·23-s + 20.3·24-s + 0.603·25-s − 2.55·26-s + 2.92·27-s − 16.5·28-s + ⋯
L(s)  = 1  − 1.93·2-s − 1.24·3-s + 2.73·4-s − 1.05·5-s + 2.40·6-s − 1.14·7-s − 3.34·8-s + 0.547·9-s + 2.04·10-s − 0.372·11-s − 3.39·12-s + 0.259·13-s + 2.21·14-s + 1.31·15-s + 3.72·16-s + 0.901·17-s − 1.05·18-s + 1.40·19-s − 2.89·20-s + 1.42·21-s + 0.720·22-s − 0.722·23-s + 4.15·24-s + 0.120·25-s − 0.501·26-s + 0.563·27-s − 3.13·28-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 \( 1 + 2.73T + 2T^{2} \)
3 \( 1 + 2.15T + 3T^{2} \)
5 \( 1 + 2.36T + 5T^{2} \)
7 \( 1 + 3.03T + 7T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 - 0.935T + 13T^{2} \)
17 \( 1 - 3.71T + 17T^{2} \)
19 \( 1 - 6.12T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 5.46T + 29T^{2} \)
31 \( 1 - 4.47T + 31T^{2} \)
37 \( 1 + 1.25T + 37T^{2} \)
41 \( 1 + 5.00T + 41T^{2} \)
43 \( 1 + 8.04T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 1.52T + 53T^{2} \)
59 \( 1 + 2.78T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 + 5.50T + 67T^{2} \)
73 \( 1 + 7.75T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 1.58T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 8.08T + 97T^{2} \)
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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

−7.64368174547166190429119211164, −6.98141420483696511734256135745, −6.28574982030495648566035656705, −5.84320514265553038980625732207, −4.92606663524495476537400991104, −3.38524302277908480650659990412, −3.18046167313570893469716680450, −1.69898743349362802874496522289, −0.65483075070968887689993378536, 0, 0.65483075070968887689993378536, 1.69898743349362802874496522289, 3.18046167313570893469716680450, 3.38524302277908480650659990412, 4.92606663524495476537400991104, 5.84320514265553038980625732207, 6.28574982030495648566035656705, 6.98141420483696511734256135745, 7.64368174547166190429119211164

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