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  − 2.45·3-s − 2.10·5-s + 3.05·9-s − 2.99·11-s + 0.922·13-s + 5.16·15-s + 6.22·17-s + 6.64·19-s − 6.34·23-s − 0.583·25-s − 0.124·27-s − 1.64·29-s + 3.00·31-s + 7.36·33-s + 9.04·37-s − 2.26·39-s − 41-s + 7.65·43-s − 6.41·45-s − 6.66·47-s − 15.3·51-s − 7.74·53-s + 6.29·55-s − 16.3·57-s − 13.3·59-s + 9.83·61-s − 1.93·65-s + ⋯
L(s)  = 1  − 1.42·3-s − 0.939·5-s + 1.01·9-s − 0.903·11-s + 0.255·13-s + 1.33·15-s + 1.50·17-s + 1.52·19-s − 1.32·23-s − 0.116·25-s − 0.0239·27-s − 0.304·29-s + 0.539·31-s + 1.28·33-s + 1.48·37-s − 0.363·39-s − 0.156·41-s + 1.16·43-s − 0.955·45-s − 0.972·47-s − 2.14·51-s − 1.06·53-s + 0.848·55-s − 2.16·57-s − 1.73·59-s + 1.25·61-s − 0.240·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6744494598$
$L(\frac12)$  $\approx$  $0.6744494598$
$L(\frac{3}{2})$   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 \( 1 + 2.45T + 3T^{2} \)
5 \( 1 + 2.10T + 5T^{2} \)
11 \( 1 + 2.99T + 11T^{2} \)
13 \( 1 - 0.922T + 13T^{2} \)
17 \( 1 - 6.22T + 17T^{2} \)
19 \( 1 - 6.64T + 19T^{2} \)
23 \( 1 + 6.34T + 23T^{2} \)
29 \( 1 + 1.64T + 29T^{2} \)
31 \( 1 - 3.00T + 31T^{2} \)
37 \( 1 - 9.04T + 37T^{2} \)
43 \( 1 - 7.65T + 43T^{2} \)
47 \( 1 + 6.66T + 47T^{2} \)
53 \( 1 + 7.74T + 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 9.83T + 61T^{2} \)
67 \( 1 + 8.26T + 67T^{2} \)
71 \( 1 + 3.01T + 71T^{2} \)
73 \( 1 + 16.5T + 73T^{2} \)
79 \( 1 - 0.193T + 79T^{2} \)
83 \( 1 - 1.85T + 83T^{2} \)
89 \( 1 - 1.02T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−7.71494894717716193707103608107, −7.33163278334833498592953530659, −6.07588396584652573299861260255, −5.91220741536010921072815672515, −5.08304931487043183270744191207, −4.48277542579392067978715939541, −3.58518191129967595136194383016, −2.84918406255436038150798141174, −1.38320120354664723478875436198, −0.47504354409650531923943355699, 0.47504354409650531923943355699, 1.38320120354664723478875436198, 2.84918406255436038150798141174, 3.58518191129967595136194383016, 4.48277542579392067978715939541, 5.08304931487043183270744191207, 5.91220741536010921072815672515, 6.07588396584652573299861260255, 7.33163278334833498592953530659, 7.71494894717716193707103608107

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