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  − 0.915·3-s − 1.31·5-s − 2.16·9-s + 2.72·11-s + 6.20·13-s + 1.19·15-s − 6.76·17-s + 5.15·19-s + 6.76·23-s − 3.28·25-s + 4.72·27-s − 6.37·29-s + 2.28·31-s − 2.49·33-s + 9.35·37-s − 5.67·39-s − 41-s − 6.74·43-s + 2.83·45-s + 8.15·47-s + 6.19·51-s − 12.5·53-s − 3.57·55-s − 4.71·57-s + 1.48·59-s − 2.36·61-s − 8.12·65-s + ⋯
L(s)  = 1  − 0.528·3-s − 0.585·5-s − 0.720·9-s + 0.822·11-s + 1.72·13-s + 0.309·15-s − 1.64·17-s + 1.18·19-s + 1.40·23-s − 0.656·25-s + 0.909·27-s − 1.18·29-s + 0.411·31-s − 0.434·33-s + 1.53·37-s − 0.909·39-s − 0.156·41-s − 1.02·43-s + 0.422·45-s + 1.18·47-s + 0.867·51-s − 1.72·53-s − 0.481·55-s − 0.625·57-s + 0.193·59-s − 0.303·61-s − 1.00·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.390810505$
$L(\frac12)$  $\approx$  $1.390810505$
$L(\frac{3}{2})$   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 \( 1 + 0.915T + 3T^{2} \)
5 \( 1 + 1.31T + 5T^{2} \)
11 \( 1 - 2.72T + 11T^{2} \)
13 \( 1 - 6.20T + 13T^{2} \)
17 \( 1 + 6.76T + 17T^{2} \)
19 \( 1 - 5.15T + 19T^{2} \)
23 \( 1 - 6.76T + 23T^{2} \)
29 \( 1 + 6.37T + 29T^{2} \)
31 \( 1 - 2.28T + 31T^{2} \)
37 \( 1 - 9.35T + 37T^{2} \)
43 \( 1 + 6.74T + 43T^{2} \)
47 \( 1 - 8.15T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 - 1.48T + 59T^{2} \)
61 \( 1 + 2.36T + 61T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 + 1.28T + 71T^{2} \)
73 \( 1 - 3.81T + 73T^{2} \)
79 \( 1 - 4.28T + 79T^{2} \)
83 \( 1 + 2.35T + 83T^{2} \)
89 \( 1 - 3.98T + 89T^{2} \)
97 \( 1 + 7.03T + 97T^{2} \)
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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

−7.88720591773302043400922043744, −6.99892499210005384210578190757, −6.39945152690651324169274816995, −5.87239319913383237689784552005, −5.06238438586256686012939792512, −4.23235677133748672325932928624, −3.59498431246669826641392155495, −2.82629446824335236757596187743, −1.55861840585172288153663749257, −0.62952167572649113325085159030, 0.62952167572649113325085159030, 1.55861840585172288153663749257, 2.82629446824335236757596187743, 3.59498431246669826641392155495, 4.23235677133748672325932928624, 5.06238438586256686012939792512, 5.87239319913383237689784552005, 6.39945152690651324169274816995, 6.99892499210005384210578190757, 7.88720591773302043400922043744

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