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  − 0.103·3-s − 3.71·5-s − 2.98·9-s − 1.25·11-s + 0.166·13-s + 0.385·15-s + 5.79·17-s − 1.05·19-s − 8.98·23-s + 8.79·25-s + 0.621·27-s − 3.21·29-s − 9.67·31-s + 0.130·33-s + 1.70·37-s − 0.0173·39-s − 41-s − 3.83·43-s + 11.1·45-s − 7.32·47-s − 0.601·51-s + 2.82·53-s + 4.67·55-s + 0.109·57-s − 9.74·59-s − 4.82·61-s − 0.619·65-s + ⋯
L(s)  = 1  − 0.0598·3-s − 1.66·5-s − 0.996·9-s − 0.379·11-s + 0.0462·13-s + 0.0995·15-s + 1.40·17-s − 0.242·19-s − 1.87·23-s + 1.75·25-s + 0.119·27-s − 0.596·29-s − 1.73·31-s + 0.0227·33-s + 0.280·37-s − 0.00277·39-s − 0.156·41-s − 0.584·43-s + 1.65·45-s − 1.06·47-s − 0.0842·51-s + 0.387·53-s + 0.630·55-s + 0.0145·57-s − 1.26·59-s − 0.617·61-s − 0.0768·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3844838218$
$L(\frac12)$  $\approx$  $0.3844838218$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + 0.103T + 3T^{2} \)
5 \( 1 + 3.71T + 5T^{2} \)
11 \( 1 + 1.25T + 11T^{2} \)
13 \( 1 - 0.166T + 13T^{2} \)
17 \( 1 - 5.79T + 17T^{2} \)
19 \( 1 + 1.05T + 19T^{2} \)
23 \( 1 + 8.98T + 23T^{2} \)
29 \( 1 + 3.21T + 29T^{2} \)
31 \( 1 + 9.67T + 31T^{2} \)
37 \( 1 - 1.70T + 37T^{2} \)
43 \( 1 + 3.83T + 43T^{2} \)
47 \( 1 + 7.32T + 47T^{2} \)
53 \( 1 - 2.82T + 53T^{2} \)
59 \( 1 + 9.74T + 59T^{2} \)
61 \( 1 + 4.82T + 61T^{2} \)
67 \( 1 + 1.21T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 - 6.00T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 4.56T + 97T^{2} \)
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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

−7.82116317769428654909205895295, −7.47279676762171337653701911886, −6.43774680246500113308280167698, −5.68980481632751172641072901020, −5.07241445704573518653685255555, −4.09251902094732645452374306934, −3.56217224113534847531331039452, −2.93674864756497881827373397853, −1.73449259948734618070751046195, −0.29762947337552829918190691215, 0.29762947337552829918190691215, 1.73449259948734618070751046195, 2.93674864756497881827373397853, 3.56217224113534847531331039452, 4.09251902094732645452374306934, 5.07241445704573518653685255555, 5.68980481632751172641072901020, 6.43774680246500113308280167698, 7.47279676762171337653701911886, 7.82116317769428654909205895295

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