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  + 0.741·3-s − 2.48·5-s − 2.45·9-s − 5.32·11-s + 4.45·13-s − 1.84·15-s − 5.22·17-s + 1.97·19-s − 6.97·23-s + 1.18·25-s − 4.04·27-s + 3.94·29-s + 4.03·31-s − 3.95·33-s + 8.44·37-s + 3.30·39-s − 41-s − 10.5·43-s + 6.09·45-s − 4.67·47-s − 3.87·51-s + 2.15·53-s + 13.2·55-s + 1.46·57-s + 3.06·59-s − 12.0·61-s − 11.0·65-s + ⋯
L(s)  = 1  + 0.428·3-s − 1.11·5-s − 0.816·9-s − 1.60·11-s + 1.23·13-s − 0.475·15-s − 1.26·17-s + 0.452·19-s − 1.45·23-s + 0.236·25-s − 0.777·27-s + 0.733·29-s + 0.725·31-s − 0.687·33-s + 1.38·37-s + 0.528·39-s − 0.156·41-s − 1.61·43-s + 0.908·45-s − 0.681·47-s − 0.542·51-s + 0.295·53-s + 1.78·55-s + 0.193·57-s + 0.399·59-s − 1.53·61-s − 1.37·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7963816954$
$L(\frac12)$  $\approx$  $0.7963816954$
$L(\frac{3}{2})$   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 \( 1 - 0.741T + 3T^{2} \)
5 \( 1 + 2.48T + 5T^{2} \)
11 \( 1 + 5.32T + 11T^{2} \)
13 \( 1 - 4.45T + 13T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 - 1.97T + 19T^{2} \)
23 \( 1 + 6.97T + 23T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 - 4.03T + 31T^{2} \)
37 \( 1 - 8.44T + 37T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 4.67T + 47T^{2} \)
53 \( 1 - 2.15T + 53T^{2} \)
59 \( 1 - 3.06T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + 0.549T + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 - 3.02T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−7.935449729103507116170436279125, −7.44652671480120668037097432808, −6.32267653930416652853625382180, −5.90461168921145358583769210614, −4.85513644851181608865733623088, −4.27965498217098345191216341384, −3.37912997338195859697619094282, −2.85251477432079314841296651364, −1.92674858046763407640214720333, −0.40745626347520622290016569799, 0.40745626347520622290016569799, 1.92674858046763407640214720333, 2.85251477432079314841296651364, 3.37912997338195859697619094282, 4.27965498217098345191216341384, 4.85513644851181608865733623088, 5.90461168921145358583769210614, 6.32267653930416652853625382180, 7.44652671480120668037097432808, 7.935449729103507116170436279125

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