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  + 2.71·3-s − 0.521·5-s + 4.38·9-s − 0.0523·11-s − 5.95·13-s − 1.41·15-s + 8.06·17-s + 4.41·19-s + 5.28·23-s − 4.72·25-s + 3.76·27-s + 4.73·29-s − 8.03·31-s − 0.142·33-s − 5.28·37-s − 16.1·39-s + 41-s + 3.65·43-s − 2.28·45-s + 4.87·47-s + 21.9·51-s − 1.13·53-s + 0.0273·55-s + 11.9·57-s + 5.21·59-s + 10.2·61-s + 3.10·65-s + ⋯
L(s)  = 1  + 1.56·3-s − 0.233·5-s + 1.46·9-s − 0.0157·11-s − 1.65·13-s − 0.365·15-s + 1.95·17-s + 1.01·19-s + 1.10·23-s − 0.945·25-s + 0.725·27-s + 0.879·29-s − 1.44·31-s − 0.0247·33-s − 0.868·37-s − 2.59·39-s + 0.156·41-s + 0.557·43-s − 0.340·45-s + 0.710·47-s + 3.06·51-s − 0.155·53-s + 0.00368·55-s + 1.58·57-s + 0.679·59-s + 1.31·61-s + 0.385·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.713922721$
$L(\frac12)$  $\approx$  $3.713922721$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - 2.71T + 3T^{2} \)
5 \( 1 + 0.521T + 5T^{2} \)
11 \( 1 + 0.0523T + 11T^{2} \)
13 \( 1 + 5.95T + 13T^{2} \)
17 \( 1 - 8.06T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
23 \( 1 - 5.28T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + 8.03T + 31T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
43 \( 1 - 3.65T + 43T^{2} \)
47 \( 1 - 4.87T + 47T^{2} \)
53 \( 1 + 1.13T + 53T^{2} \)
59 \( 1 - 5.21T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 9.37T + 67T^{2} \)
71 \( 1 - 9.25T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 - 6.66T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 8.41T + 97T^{2} \)
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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

−7.81014737211440921155303921989, −7.39084917346633678077108153035, −6.85668673910400211265866713501, −5.42176004392672589629142542149, −5.15932638587133987229314416805, −3.95154508392668501063075337046, −3.42443359465262184030506067624, −2.74782839714979358743777684625, −2.02988123158199272133510944566, −0.894181869907808049522342404079, 0.894181869907808049522342404079, 2.02988123158199272133510944566, 2.74782839714979358743777684625, 3.42443359465262184030506067624, 3.95154508392668501063075337046, 5.15932638587133987229314416805, 5.42176004392672589629142542149, 6.85668673910400211265866713501, 7.39084917346633678077108153035, 7.81014737211440921155303921989

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