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  − 0.146·3-s + 3.56·5-s − 2.97·9-s − 5.58·11-s − 6.25·13-s − 0.522·15-s − 6.24·17-s + 7.23·19-s + 5.27·23-s + 7.73·25-s + 0.875·27-s − 4.42·29-s + 1.06·31-s + 0.817·33-s + 2.40·37-s + 0.917·39-s − 41-s + 12.2·43-s − 10.6·45-s − 4.35·47-s + 0.915·51-s + 9.39·53-s − 19.9·55-s − 1.05·57-s − 13.2·59-s + 14.3·61-s − 22.3·65-s + ⋯
L(s)  = 1  − 0.0845·3-s + 1.59·5-s − 0.992·9-s − 1.68·11-s − 1.73·13-s − 0.135·15-s − 1.51·17-s + 1.65·19-s + 1.10·23-s + 1.54·25-s + 0.168·27-s − 0.821·29-s + 0.190·31-s + 0.142·33-s + 0.395·37-s + 0.146·39-s − 0.156·41-s + 1.87·43-s − 1.58·45-s − 0.635·47-s + 0.128·51-s + 1.29·53-s − 2.68·55-s − 0.140·57-s − 1.73·59-s + 1.83·61-s − 2.77·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.715894842$
$L(\frac12)$  $\approx$  $1.715894842$
$L(\frac{3}{2})$   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 \( 1 + 0.146T + 3T^{2} \)
5 \( 1 - 3.56T + 5T^{2} \)
11 \( 1 + 5.58T + 11T^{2} \)
13 \( 1 + 6.25T + 13T^{2} \)
17 \( 1 + 6.24T + 17T^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
23 \( 1 - 5.27T + 23T^{2} \)
29 \( 1 + 4.42T + 29T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 - 2.40T + 37T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 4.35T + 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 + 0.0228T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 1.46T + 79T^{2} \)
83 \( 1 - 4.21T + 83T^{2} \)
89 \( 1 + 3.56T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−7.65377180920620246934328554999, −7.22544862728090793326816064901, −6.33891596302945409226456512795, −5.52803757026844189912906070211, −5.23837424791495890079491038546, −4.68424382782600430657916523229, −3.11484608489819833308808827615, −2.49503686934105726620473334755, −2.15268724227087274290985271863, −0.60855407371942997857355931442, 0.60855407371942997857355931442, 2.15268724227087274290985271863, 2.49503686934105726620473334755, 3.11484608489819833308808827615, 4.68424382782600430657916523229, 5.23837424791495890079491038546, 5.52803757026844189912906070211, 6.33891596302945409226456512795, 7.22544862728090793326816064901, 7.65377180920620246934328554999

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