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  − 3.15·3-s + 2.22·5-s + 6.96·9-s − 4.12·11-s + 2.30·13-s − 7.00·15-s − 2.24·17-s + 1.92·19-s − 2.59·23-s − 0.0707·25-s − 12.5·27-s + 4.54·29-s + 0.881·31-s + 13.0·33-s + 7.08·37-s − 7.29·39-s − 41-s − 12.1·43-s + 15.4·45-s − 5.83·47-s + 7.09·51-s − 8.01·53-s − 9.15·55-s − 6.07·57-s + 11.7·59-s + 7.46·61-s + 5.12·65-s + ⋯
L(s)  = 1  − 1.82·3-s + 0.992·5-s + 2.32·9-s − 1.24·11-s + 0.640·13-s − 1.80·15-s − 0.545·17-s + 0.441·19-s − 0.541·23-s − 0.0141·25-s − 2.41·27-s + 0.844·29-s + 0.158·31-s + 2.26·33-s + 1.16·37-s − 1.16·39-s − 0.156·41-s − 1.84·43-s + 2.30·45-s − 0.850·47-s + 0.993·51-s − 1.10·53-s − 1.23·55-s − 0.804·57-s + 1.52·59-s + 0.955·61-s + 0.636·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9909057493$
$L(\frac12)$  $\approx$  $0.9909057493$
$L(\frac{3}{2})$   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 \( 1 + 3.15T + 3T^{2} \)
5 \( 1 - 2.22T + 5T^{2} \)
11 \( 1 + 4.12T + 11T^{2} \)
13 \( 1 - 2.30T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 - 1.92T + 19T^{2} \)
23 \( 1 + 2.59T + 23T^{2} \)
29 \( 1 - 4.54T + 29T^{2} \)
31 \( 1 - 0.881T + 31T^{2} \)
37 \( 1 - 7.08T + 37T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + 5.83T + 47T^{2} \)
53 \( 1 + 8.01T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 7.46T + 61T^{2} \)
67 \( 1 - 1.58T + 67T^{2} \)
71 \( 1 - 8.75T + 71T^{2} \)
73 \( 1 - 2.18T + 73T^{2} \)
79 \( 1 + 0.403T + 79T^{2} \)
83 \( 1 - 5.26T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 3.49T + 97T^{2} \)
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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

−7.75425580621249342500531828502, −6.70770860922065067955378209839, −6.40777012451440833722789688790, −5.73219232895437186331061904487, −5.15414179427919916216171903998, −4.71837231466934828786941111536, −3.66617565607766099271814474895, −2.44570811034960381317576534209, −1.57574216860640521640354406639, −0.55556566917000263484103601342, 0.55556566917000263484103601342, 1.57574216860640521640354406639, 2.44570811034960381317576534209, 3.66617565607766099271814474895, 4.71837231466934828786941111536, 5.15414179427919916216171903998, 5.73219232895437186331061904487, 6.40777012451440833722789688790, 6.70770860922065067955378209839, 7.75425580621249342500531828502

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