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  + 2.73·3-s − 0.863·5-s + 4.49·9-s − 5.59·11-s − 1.90·13-s − 2.36·15-s − 4.01·17-s − 1.91·19-s + 9.06·23-s − 4.25·25-s + 4.08·27-s + 5.61·29-s + 9.11·31-s − 15.3·33-s + 1.45·37-s − 5.20·39-s − 41-s + 7.83·43-s − 3.87·45-s + 10.1·47-s − 11.0·51-s − 1.16·53-s + 4.82·55-s − 5.24·57-s + 6.86·59-s + 5.22·61-s + 1.63·65-s + ⋯
L(s)  = 1  + 1.58·3-s − 0.385·5-s + 1.49·9-s − 1.68·11-s − 0.526·13-s − 0.610·15-s − 0.974·17-s − 0.439·19-s + 1.89·23-s − 0.851·25-s + 0.786·27-s + 1.04·29-s + 1.63·31-s − 2.66·33-s + 0.239·37-s − 0.832·39-s − 0.156·41-s + 1.19·43-s − 0.578·45-s + 1.48·47-s − 1.54·51-s − 0.159·53-s + 0.650·55-s − 0.695·57-s + 0.894·59-s + 0.669·61-s + 0.203·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.921970689$
$L(\frac12)$  $\approx$  $2.921970689$
$L(\frac{3}{2})$   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 \( 1 - 2.73T + 3T^{2} \)
5 \( 1 + 0.863T + 5T^{2} \)
11 \( 1 + 5.59T + 11T^{2} \)
13 \( 1 + 1.90T + 13T^{2} \)
17 \( 1 + 4.01T + 17T^{2} \)
19 \( 1 + 1.91T + 19T^{2} \)
23 \( 1 - 9.06T + 23T^{2} \)
29 \( 1 - 5.61T + 29T^{2} \)
31 \( 1 - 9.11T + 31T^{2} \)
37 \( 1 - 1.45T + 37T^{2} \)
43 \( 1 - 7.83T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 1.16T + 53T^{2} \)
59 \( 1 - 6.86T + 59T^{2} \)
61 \( 1 - 5.22T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 - 3.14T + 71T^{2} \)
73 \( 1 + 4.41T + 73T^{2} \)
79 \( 1 + 5.30T + 79T^{2} \)
83 \( 1 - 6.73T + 83T^{2} \)
89 \( 1 + 3.99T + 89T^{2} \)
97 \( 1 - 3.44T + 97T^{2} \)
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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

−8.074596734448355516585794569958, −7.27327036546984840096532968690, −6.81758396735494991176038686233, −5.66595208532198792360029394084, −4.76392034438519258807073107261, −4.27819300358258753741598005478, −3.28698904595410456377036817255, −2.52120252384637867897250645128, −2.33927596914012388835367660247, −0.75223864219943322258050074137, 0.75223864219943322258050074137, 2.33927596914012388835367660247, 2.52120252384637867897250645128, 3.28698904595410456377036817255, 4.27819300358258753741598005478, 4.76392034438519258807073107261, 5.66595208532198792360029394084, 6.81758396735494991176038686233, 7.27327036546984840096532968690, 8.074596734448355516585794569958

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