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  + 2.58·3-s + 3.78·5-s + 3.69·9-s − 3.59·11-s + 3.80·13-s + 9.79·15-s + 4.90·17-s + 0.726·19-s + 2.67·23-s + 9.32·25-s + 1.80·27-s − 1.31·29-s + 4.60·31-s − 9.30·33-s + 6.36·37-s + 9.84·39-s + 41-s − 6.79·43-s + 13.9·45-s − 9.58·47-s + 12.6·51-s + 3.00·53-s − 13.6·55-s + 1.88·57-s + 4.98·59-s − 2.28·61-s + 14.4·65-s + ⋯
L(s)  = 1  + 1.49·3-s + 1.69·5-s + 1.23·9-s − 1.08·11-s + 1.05·13-s + 2.52·15-s + 1.18·17-s + 0.166·19-s + 0.558·23-s + 1.86·25-s + 0.347·27-s − 0.244·29-s + 0.826·31-s − 1.62·33-s + 1.04·37-s + 1.57·39-s + 0.156·41-s − 1.03·43-s + 2.08·45-s − 1.39·47-s + 1.77·51-s + 0.412·53-s − 1.83·55-s + 0.249·57-s + 0.649·59-s − 0.292·61-s + 1.78·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.586399824$
$L(\frac12)$  $\approx$  $5.586399824$
$L(\frac{3}{2})$   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 \( 1 - 2.58T + 3T^{2} \)
5 \( 1 - 3.78T + 5T^{2} \)
11 \( 1 + 3.59T + 11T^{2} \)
13 \( 1 - 3.80T + 13T^{2} \)
17 \( 1 - 4.90T + 17T^{2} \)
19 \( 1 - 0.726T + 19T^{2} \)
23 \( 1 - 2.67T + 23T^{2} \)
29 \( 1 + 1.31T + 29T^{2} \)
31 \( 1 - 4.60T + 31T^{2} \)
37 \( 1 - 6.36T + 37T^{2} \)
43 \( 1 + 6.79T + 43T^{2} \)
47 \( 1 + 9.58T + 47T^{2} \)
53 \( 1 - 3.00T + 53T^{2} \)
59 \( 1 - 4.98T + 59T^{2} \)
61 \( 1 + 2.28T + 61T^{2} \)
67 \( 1 + 5.58T + 67T^{2} \)
71 \( 1 + 1.69T + 71T^{2} \)
73 \( 1 - 0.259T + 73T^{2} \)
79 \( 1 - 2.89T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 - 1.39T + 97T^{2} \)
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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

−8.134634340061803247265121104682, −7.24683115900816840457081319065, −6.44384819206602247338715480007, −5.69024700832968233871088946509, −5.20011664096717482433793364660, −4.14911139354141184049504729877, −3.02006217546588019828588410278, −2.87877845477081835149839321406, −1.87372683082080326141060140420, −1.21009869364004268598471535734, 1.21009869364004268598471535734, 1.87372683082080326141060140420, 2.87877845477081835149839321406, 3.02006217546588019828588410278, 4.14911139354141184049504729877, 5.20011664096717482433793364660, 5.69024700832968233871088946509, 6.44384819206602247338715480007, 7.24683115900816840457081319065, 8.134634340061803247265121104682

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