Properties

Degree 2
Conductor $ 2^{2} \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.10·3-s − 3.31·5-s + 6.61·9-s − 0.792·11-s + 6.10·13-s + 10.2·15-s + 0.423·17-s + 0.177·19-s + 5.55·23-s + 5.98·25-s − 11.2·27-s + 6.87·29-s + 1.03·31-s + 2.45·33-s + 10.9·37-s − 18.9·39-s − 41-s + 7.60·43-s − 21.9·45-s − 0.719·47-s − 1.31·51-s + 9.70·53-s + 2.62·55-s − 0.551·57-s + 8.31·59-s + 9.24·61-s − 20.2·65-s + ⋯
L(s)  = 1  − 1.79·3-s − 1.48·5-s + 2.20·9-s − 0.238·11-s + 1.69·13-s + 2.65·15-s + 0.102·17-s + 0.0407·19-s + 1.15·23-s + 1.19·25-s − 2.15·27-s + 1.27·29-s + 0.186·31-s + 0.427·33-s + 1.79·37-s − 3.02·39-s − 0.156·41-s + 1.15·43-s − 3.26·45-s − 0.104·47-s − 0.183·51-s + 1.33·53-s + 0.353·55-s − 0.0730·57-s + 1.08·59-s + 1.18·61-s − 2.50·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9429738925$
$L(\frac12)$  $\approx$  $0.9429738925$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + 3.10T + 3T^{2} \)
5 \( 1 + 3.31T + 5T^{2} \)
11 \( 1 + 0.792T + 11T^{2} \)
13 \( 1 - 6.10T + 13T^{2} \)
17 \( 1 - 0.423T + 17T^{2} \)
19 \( 1 - 0.177T + 19T^{2} \)
23 \( 1 - 5.55T + 23T^{2} \)
29 \( 1 - 6.87T + 29T^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
43 \( 1 - 7.60T + 43T^{2} \)
47 \( 1 + 0.719T + 47T^{2} \)
53 \( 1 - 9.70T + 53T^{2} \)
59 \( 1 - 8.31T + 59T^{2} \)
61 \( 1 - 9.24T + 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 - 6.50T + 71T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 - 0.0105T + 79T^{2} \)
83 \( 1 - 7.04T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.77744425984653796420283679843, −6.89880707938082066533827084511, −6.53210262434017508034476586063, −5.71160432520846383040008160811, −5.11605388657757858964479463875, −4.22003750036715995102745283201, −3.94084384783357204518374548954, −2.82015063396702658128137698433, −1.10473739950142081428321693002, −0.68365212194153441821987358470, 0.68365212194153441821987358470, 1.10473739950142081428321693002, 2.82015063396702658128137698433, 3.94084384783357204518374548954, 4.22003750036715995102745283201, 5.11605388657757858964479463875, 5.71160432520846383040008160811, 6.53210262434017508034476586063, 6.89880707938082066533827084511, 7.77744425984653796420283679843

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