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  + 0.476·3-s − 2.17·5-s − 2.77·9-s − 0.849·11-s + 0.662·13-s − 1.03·15-s − 5.15·17-s − 6.79·19-s − 0.144·23-s − 0.259·25-s − 2.74·27-s − 9.46·29-s + 8.50·31-s − 0.404·33-s − 7.75·37-s + 0.315·39-s − 41-s − 2.77·43-s + 6.03·45-s + 3.45·47-s − 2.45·51-s + 7.28·53-s + 1.85·55-s − 3.23·57-s + 9.84·59-s + 9.68·61-s − 1.44·65-s + ⋯
L(s)  = 1  + 0.274·3-s − 0.973·5-s − 0.924·9-s − 0.256·11-s + 0.183·13-s − 0.267·15-s − 1.25·17-s − 1.55·19-s − 0.0301·23-s − 0.0518·25-s − 0.528·27-s − 1.75·29-s + 1.52·31-s − 0.0704·33-s − 1.27·37-s + 0.0504·39-s − 0.156·41-s − 0.422·43-s + 0.900·45-s + 0.504·47-s − 0.343·51-s + 1.00·53-s + 0.249·55-s − 0.428·57-s + 1.28·59-s + 1.23·61-s − 0.178·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6584989810$
$L(\frac12)$  $\approx$  $0.6584989810$
$L(\frac{3}{2})$   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 \( 1 - 0.476T + 3T^{2} \)
5 \( 1 + 2.17T + 5T^{2} \)
11 \( 1 + 0.849T + 11T^{2} \)
13 \( 1 - 0.662T + 13T^{2} \)
17 \( 1 + 5.15T + 17T^{2} \)
19 \( 1 + 6.79T + 19T^{2} \)
23 \( 1 + 0.144T + 23T^{2} \)
29 \( 1 + 9.46T + 29T^{2} \)
31 \( 1 - 8.50T + 31T^{2} \)
37 \( 1 + 7.75T + 37T^{2} \)
43 \( 1 + 2.77T + 43T^{2} \)
47 \( 1 - 3.45T + 47T^{2} \)
53 \( 1 - 7.28T + 53T^{2} \)
59 \( 1 - 9.84T + 59T^{2} \)
61 \( 1 - 9.68T + 61T^{2} \)
67 \( 1 - 2.94T + 67T^{2} \)
71 \( 1 - 3.88T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 - 5.63T + 79T^{2} \)
83 \( 1 - 8.76T + 83T^{2} \)
89 \( 1 + 5.94T + 89T^{2} \)
97 \( 1 - 2.25T + 97T^{2} \)
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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

−8.011324282432508514460512031144, −7.14424414480618562801396985493, −6.55901834853862673921369650120, −5.76288069704130459598975487319, −4.97955406241519360875964497577, −4.06363016459030620119526648571, −3.70075633018236602158977456899, −2.61357634979203658088589946779, −1.99740179631202251100924518413, −0.36952656458875077245201905468, 0.36952656458875077245201905468, 1.99740179631202251100924518413, 2.61357634979203658088589946779, 3.70075633018236602158977456899, 4.06363016459030620119526648571, 4.97955406241519360875964497577, 5.76288069704130459598975487319, 6.55901834853862673921369650120, 7.14424414480618562801396985493, 8.011324282432508514460512031144

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