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  − 1.37·3-s + 3.83·5-s − 1.10·9-s + 0.294·11-s + 2.83·13-s − 5.27·15-s + 3.23·17-s − 5.79·19-s + 5.90·23-s + 9.69·25-s + 5.65·27-s + 3.65·29-s − 1.59·31-s − 0.405·33-s + 5.20·37-s − 3.90·39-s − 41-s + 5.12·43-s − 4.22·45-s − 12.1·47-s − 4.46·51-s − 7.84·53-s + 1.12·55-s + 7.97·57-s + 2.50·59-s + 2.24·61-s + 10.8·65-s + ⋯
L(s)  = 1  − 0.795·3-s + 1.71·5-s − 0.367·9-s + 0.0888·11-s + 0.786·13-s − 1.36·15-s + 0.785·17-s − 1.32·19-s + 1.23·23-s + 1.93·25-s + 1.08·27-s + 0.678·29-s − 0.285·31-s − 0.0706·33-s + 0.856·37-s − 0.625·39-s − 0.156·41-s + 0.780·43-s − 0.630·45-s − 1.77·47-s − 0.624·51-s − 1.07·53-s + 0.152·55-s + 1.05·57-s + 0.326·59-s + 0.287·61-s + 1.34·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.283771397$
$L(\frac12)$  $\approx$  $2.283771397$
$L(\frac{3}{2})$   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 \( 1 + 1.37T + 3T^{2} \)
5 \( 1 - 3.83T + 5T^{2} \)
11 \( 1 - 0.294T + 11T^{2} \)
13 \( 1 - 2.83T + 13T^{2} \)
17 \( 1 - 3.23T + 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 1.59T + 31T^{2} \)
37 \( 1 - 5.20T + 37T^{2} \)
43 \( 1 - 5.12T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 7.84T + 53T^{2} \)
59 \( 1 - 2.50T + 59T^{2} \)
61 \( 1 - 2.24T + 61T^{2} \)
67 \( 1 + 8.73T + 67T^{2} \)
71 \( 1 + 1.66T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 + 0.253T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 3.82T + 89T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−7.87871669637833568664806519150, −6.68412019952691965941502135174, −6.36797870545002444943623153586, −5.85001470417201376970910790186, −5.18631769323716463754982158065, −4.61469260074401695064163641818, −3.36474013295306986083437972389, −2.58859036004748523636816895424, −1.66864174701634347633717804200, −0.814747744931286547055860993312, 0.814747744931286547055860993312, 1.66864174701634347633717804200, 2.58859036004748523636816895424, 3.36474013295306986083437972389, 4.61469260074401695064163641818, 5.18631769323716463754982158065, 5.85001470417201376970910790186, 6.36797870545002444943623153586, 6.68412019952691965941502135174, 7.87871669637833568664806519150

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