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  − 0.0954·3-s − 0.581·5-s − 2.99·9-s − 4.59·11-s − 5.44·13-s + 0.0554·15-s − 0.983·17-s + 0.599·19-s + 5.72·23-s − 4.66·25-s + 0.571·27-s + 6.69·29-s + 1.57·31-s + 0.438·33-s − 8.89·37-s + 0.519·39-s − 41-s − 5.24·43-s + 1.73·45-s − 7.66·47-s + 0.0938·51-s − 11.3·53-s + 2.66·55-s − 0.0572·57-s + 7.53·59-s + 5.90·61-s + 3.16·65-s + ⋯
L(s)  = 1  − 0.0550·3-s − 0.259·5-s − 0.996·9-s − 1.38·11-s − 1.51·13-s + 0.0143·15-s − 0.238·17-s + 0.137·19-s + 1.19·23-s − 0.932·25-s + 0.110·27-s + 1.24·29-s + 0.283·31-s + 0.0762·33-s − 1.46·37-s + 0.0832·39-s − 0.156·41-s − 0.799·43-s + 0.259·45-s − 1.11·47-s + 0.0131·51-s − 1.55·53-s + 0.359·55-s − 0.00758·57-s + 0.980·59-s + 0.755·61-s + 0.392·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6123998613$
$L(\frac12)$  $\approx$  $0.6123998613$
$L(\frac{3}{2})$   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 \( 1 + 0.0954T + 3T^{2} \)
5 \( 1 + 0.581T + 5T^{2} \)
11 \( 1 + 4.59T + 11T^{2} \)
13 \( 1 + 5.44T + 13T^{2} \)
17 \( 1 + 0.983T + 17T^{2} \)
19 \( 1 - 0.599T + 19T^{2} \)
23 \( 1 - 5.72T + 23T^{2} \)
29 \( 1 - 6.69T + 29T^{2} \)
31 \( 1 - 1.57T + 31T^{2} \)
37 \( 1 + 8.89T + 37T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 - 7.53T + 59T^{2} \)
61 \( 1 - 5.90T + 61T^{2} \)
67 \( 1 - 2.45T + 67T^{2} \)
71 \( 1 + 8.01T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 - 1.44T + 97T^{2} \)
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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

−7.945297023190132210941362505631, −7.14922111252467631052199215501, −6.57006288553514176971124892132, −5.52185229862619635051990862026, −5.09927947872366767473327366525, −4.51632642511709227991737768049, −3.19995102075144126385400170730, −2.82787637262431026122440533936, −1.92835007949537262955456323689, −0.36168034830975287574110869452, 0.36168034830975287574110869452, 1.92835007949537262955456323689, 2.82787637262431026122440533936, 3.19995102075144126385400170730, 4.51632642511709227991737768049, 5.09927947872366767473327366525, 5.52185229862619635051990862026, 6.57006288553514176971124892132, 7.14922111252467631052199215501, 7.945297023190132210941362505631

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