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  + 2.15·3-s − 3.75·5-s + 1.62·9-s + 4.89·11-s + 1.80·13-s − 8.07·15-s + 4.27·17-s + 5.31·19-s + 0.795·23-s + 9.10·25-s − 2.95·27-s + 0.466·29-s − 2.71·31-s + 10.5·33-s − 10.1·37-s + 3.87·39-s − 41-s − 3.98·43-s − 6.11·45-s + 2.29·47-s + 9.19·51-s − 1.15·53-s − 18.3·55-s + 11.4·57-s + 6.87·59-s + 8.97·61-s − 6.76·65-s + ⋯
L(s)  = 1  + 1.24·3-s − 1.67·5-s + 0.542·9-s + 1.47·11-s + 0.499·13-s − 2.08·15-s + 1.03·17-s + 1.21·19-s + 0.165·23-s + 1.82·25-s − 0.568·27-s + 0.0865·29-s − 0.487·31-s + 1.83·33-s − 1.66·37-s + 0.620·39-s − 0.156·41-s − 0.608·43-s − 0.911·45-s + 0.335·47-s + 1.28·51-s − 0.158·53-s − 2.47·55-s + 1.51·57-s + 0.894·59-s + 1.14·61-s − 0.839·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.722658692$
$L(\frac12)$  $\approx$  $2.722658692$
$L(\frac{3}{2})$   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 \( 1 - 2.15T + 3T^{2} \)
5 \( 1 + 3.75T + 5T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 1.80T + 13T^{2} \)
17 \( 1 - 4.27T + 17T^{2} \)
19 \( 1 - 5.31T + 19T^{2} \)
23 \( 1 - 0.795T + 23T^{2} \)
29 \( 1 - 0.466T + 29T^{2} \)
31 \( 1 + 2.71T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
43 \( 1 + 3.98T + 43T^{2} \)
47 \( 1 - 2.29T + 47T^{2} \)
53 \( 1 + 1.15T + 53T^{2} \)
59 \( 1 - 6.87T + 59T^{2} \)
61 \( 1 - 8.97T + 61T^{2} \)
67 \( 1 + 3.33T + 67T^{2} \)
71 \( 1 - 0.810T + 71T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 + 6.00T + 79T^{2} \)
83 \( 1 + 4.93T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−7.83373213810021571377927310047, −7.33730242074233230023117010296, −6.81390863909949922526729298578, −5.73088013920283929815370170095, −4.80966072024346413699447768053, −3.83765597855077756300588903075, −3.55672187130205757052512091773, −3.09063368762476464123836590911, −1.74662331036333516192599014268, −0.803020480788443876137356714188, 0.803020480788443876137356714188, 1.74662331036333516192599014268, 3.09063368762476464123836590911, 3.55672187130205757052512091773, 3.83765597855077756300588903075, 4.80966072024346413699447768053, 5.73088013920283929815370170095, 6.81390863909949922526729298578, 7.33730242074233230023117010296, 7.83373213810021571377927310047

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