Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.14·3-s + 4-s + 5-s + 2.14·6-s − 4.86·7-s + 8-s + 1.60·9-s + 10-s − 11-s + 2.14·12-s + 0.772·13-s − 4.86·14-s + 2.14·15-s + 16-s + 3.64·17-s + 1.60·18-s + 6.17·19-s + 20-s − 10.4·21-s − 22-s + 1.22·23-s + 2.14·24-s + 25-s + 0.772·26-s − 2.99·27-s − 4.86·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.23·3-s + 0.5·4-s + 0.447·5-s + 0.875·6-s − 1.83·7-s + 0.353·8-s + 0.534·9-s + 0.316·10-s − 0.301·11-s + 0.619·12-s + 0.214·13-s − 1.30·14-s + 0.553·15-s + 0.250·16-s + 0.884·17-s + 0.377·18-s + 1.41·19-s + 0.223·20-s − 2.27·21-s − 0.213·22-s + 0.255·23-s + 0.437·24-s + 0.200·25-s + 0.151·26-s − 0.576·27-s − 0.919·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4730} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.564925339$
$L(\frac12)$  $\approx$  $4.564925339$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
43 \( 1 - T \)
good3 \( 1 - 2.14T + 3T^{2} \)
7 \( 1 + 4.86T + 7T^{2} \)
13 \( 1 - 0.772T + 13T^{2} \)
17 \( 1 - 3.64T + 17T^{2} \)
19 \( 1 - 6.17T + 19T^{2} \)
23 \( 1 - 1.22T + 23T^{2} \)
29 \( 1 - 8.75T + 29T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 - 1.71T + 41T^{2} \)
47 \( 1 + 0.956T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 1.28T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 6.58T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 2.42T + 79T^{2} \)
83 \( 1 + 4.52T + 83T^{2} \)
89 \( 1 + 3.19T + 89T^{2} \)
97 \( 1 + 9.25T + 97T^{2} \)
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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

−8.242844369647639682509710433164, −7.50199915953742004725418158140, −6.85531699248844793936845521427, −6.01100406700725422726084026576, −5.53738619901225416889952703516, −4.35887706200938652015084585398, −3.41877117434027007029548890128, −3.00917912685283244856454740530, −2.48481097891978531670640045817, −1.02375406043033238903659347373, 1.02375406043033238903659347373, 2.48481097891978531670640045817, 3.00917912685283244856454740530, 3.41877117434027007029548890128, 4.35887706200938652015084585398, 5.53738619901225416889952703516, 6.01100406700725422726084026576, 6.85531699248844793936845521427, 7.50199915953742004725418158140, 8.242844369647639682509710433164

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