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  − 0.284·3-s + 2.38·5-s − 2.91·9-s + 5.72·11-s + 5.63·13-s − 0.680·15-s + 0.225·17-s − 1.88·19-s + 1.32·23-s + 0.708·25-s + 1.68·27-s + 0.835·29-s − 4.68·31-s − 1.63·33-s + 5.80·37-s − 1.60·39-s + 41-s − 0.313·43-s − 6.97·45-s − 0.467·47-s − 0.0641·51-s + 11.1·53-s + 13.6·55-s + 0.536·57-s + 7.92·59-s − 4.62·61-s + 13.4·65-s + ⋯
L(s)  = 1  − 0.164·3-s + 1.06·5-s − 0.972·9-s + 1.72·11-s + 1.56·13-s − 0.175·15-s + 0.0546·17-s − 0.432·19-s + 0.275·23-s + 0.141·25-s + 0.324·27-s + 0.155·29-s − 0.840·31-s − 0.283·33-s + 0.954·37-s − 0.257·39-s + 0.156·41-s − 0.0478·43-s − 1.03·45-s − 0.0681·47-s − 0.00898·51-s + 1.53·53-s + 1.84·55-s + 0.0710·57-s + 1.03·59-s − 0.591·61-s + 1.67·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.843517626$
$L(\frac12)$  $\approx$  $2.843517626$
$L(\frac{3}{2})$   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 \( 1 + 0.284T + 3T^{2} \)
5 \( 1 - 2.38T + 5T^{2} \)
11 \( 1 - 5.72T + 11T^{2} \)
13 \( 1 - 5.63T + 13T^{2} \)
17 \( 1 - 0.225T + 17T^{2} \)
19 \( 1 + 1.88T + 19T^{2} \)
23 \( 1 - 1.32T + 23T^{2} \)
29 \( 1 - 0.835T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 - 5.80T + 37T^{2} \)
43 \( 1 + 0.313T + 43T^{2} \)
47 \( 1 + 0.467T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 7.92T + 59T^{2} \)
61 \( 1 + 4.62T + 61T^{2} \)
67 \( 1 - 0.187T + 67T^{2} \)
71 \( 1 - 2.39T + 71T^{2} \)
73 \( 1 - 6.57T + 73T^{2} \)
79 \( 1 + 3.56T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 - 8.14T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−7.983359308018514198664192581781, −6.78762243514778917552566753710, −6.40457228183048386014667426528, −5.83489866690500568481593198707, −5.30544353345127781639871229528, −4.11305036793505800645496486770, −3.62967686322834269937807995660, −2.59092530608944814438035804628, −1.68116678599495515567825494601, −0.901786839796998459657247317865, 0.901786839796998459657247317865, 1.68116678599495515567825494601, 2.59092530608944814438035804628, 3.62967686322834269937807995660, 4.11305036793505800645496486770, 5.30544353345127781639871229528, 5.83489866690500568481593198707, 6.40457228183048386014667426528, 6.78762243514778917552566753710, 7.983359308018514198664192581781

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