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  − 2.96·3-s + 4.34·5-s + 5.77·9-s − 0.777·11-s − 0.473·13-s − 12.8·15-s + 1.81·17-s + 6.26·19-s + 5.33·23-s + 13.8·25-s − 8.21·27-s + 4.40·29-s − 5.82·31-s + 2.30·33-s − 8.34·37-s + 1.40·39-s + 41-s − 6.18·43-s + 25.0·45-s + 6.01·47-s − 5.36·51-s + 11.5·53-s − 3.37·55-s − 18.5·57-s − 0.805·59-s + 11.1·61-s − 2.05·65-s + ⋯
L(s)  = 1  − 1.70·3-s + 1.94·5-s + 1.92·9-s − 0.234·11-s − 0.131·13-s − 3.32·15-s + 0.439·17-s + 1.43·19-s + 1.11·23-s + 2.77·25-s − 1.58·27-s + 0.818·29-s − 1.04·31-s + 0.400·33-s − 1.37·37-s + 0.224·39-s + 0.156·41-s − 0.942·43-s + 3.73·45-s + 0.877·47-s − 0.751·51-s + 1.59·53-s − 0.455·55-s − 2.45·57-s − 0.104·59-s + 1.42·61-s − 0.255·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.922027202$
$L(\frac12)$  $\approx$  $1.922027202$
$L(\frac{3}{2})$   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 \( 1 + 2.96T + 3T^{2} \)
5 \( 1 - 4.34T + 5T^{2} \)
11 \( 1 + 0.777T + 11T^{2} \)
13 \( 1 + 0.473T + 13T^{2} \)
17 \( 1 - 1.81T + 17T^{2} \)
19 \( 1 - 6.26T + 19T^{2} \)
23 \( 1 - 5.33T + 23T^{2} \)
29 \( 1 - 4.40T + 29T^{2} \)
31 \( 1 + 5.82T + 31T^{2} \)
37 \( 1 + 8.34T + 37T^{2} \)
43 \( 1 + 6.18T + 43T^{2} \)
47 \( 1 - 6.01T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 0.805T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 1.94T + 71T^{2} \)
73 \( 1 - 0.839T + 73T^{2} \)
79 \( 1 - 7.38T + 79T^{2} \)
83 \( 1 - 9.38T + 83T^{2} \)
89 \( 1 + 1.70T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−7.40974964593754864080874391471, −6.93992095772863534796156403210, −6.28554123173001889433390668373, −5.64172817742191527206267184405, −5.19057374556804852661555678075, −4.90690369938854899586375063992, −3.49970762432093334516101517118, −2.47637288781717771262369656558, −1.48255044269548766941303595874, −0.828744725805483586060587084912, 0.828744725805483586060587084912, 1.48255044269548766941303595874, 2.47637288781717771262369656558, 3.49970762432093334516101517118, 4.90690369938854899586375063992, 5.19057374556804852661555678075, 5.64172817742191527206267184405, 6.28554123173001889433390668373, 6.93992095772863534796156403210, 7.40974964593754864080874391471

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