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  + 3.24·3-s + 1.49·5-s + 7.51·9-s − 2.60·11-s + 2.15·13-s + 4.85·15-s + 7.81·17-s − 4.98·19-s − 5.00·23-s − 2.75·25-s + 14.6·27-s + 0.404·29-s + 1.13·31-s − 8.43·33-s − 1.67·37-s + 7.00·39-s − 41-s − 0.0986·43-s + 11.2·45-s + 6.72·47-s + 25.3·51-s + 13.0·53-s − 3.89·55-s − 16.1·57-s + 15.1·59-s − 1.87·61-s + 3.23·65-s + ⋯
L(s)  = 1  + 1.87·3-s + 0.669·5-s + 2.50·9-s − 0.784·11-s + 0.598·13-s + 1.25·15-s + 1.89·17-s − 1.14·19-s − 1.04·23-s − 0.551·25-s + 2.81·27-s + 0.0750·29-s + 0.203·31-s − 1.46·33-s − 0.275·37-s + 1.12·39-s − 0.156·41-s − 0.0150·43-s + 1.67·45-s + 0.980·47-s + 3.54·51-s + 1.78·53-s − 0.524·55-s − 2.14·57-s + 1.97·59-s − 0.239·61-s + 0.400·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.287420967$
$L(\frac12)$  $\approx$  $5.287420967$
$L(\frac{3}{2})$   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 \( 1 - 3.24T + 3T^{2} \)
5 \( 1 - 1.49T + 5T^{2} \)
11 \( 1 + 2.60T + 11T^{2} \)
13 \( 1 - 2.15T + 13T^{2} \)
17 \( 1 - 7.81T + 17T^{2} \)
19 \( 1 + 4.98T + 19T^{2} \)
23 \( 1 + 5.00T + 23T^{2} \)
29 \( 1 - 0.404T + 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 + 1.67T + 37T^{2} \)
43 \( 1 + 0.0986T + 43T^{2} \)
47 \( 1 - 6.72T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 - 15.1T + 59T^{2} \)
61 \( 1 + 1.87T + 61T^{2} \)
67 \( 1 + 4.60T + 67T^{2} \)
71 \( 1 + 1.21T + 71T^{2} \)
73 \( 1 - 3.16T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−7.80485806116074002038213925499, −7.57130856174502142911339740771, −6.51643178541469402322785891964, −5.79058635469948905459477007171, −4.97263177724634349634715496710, −3.82699059385246898579623431730, −3.60830230138185988586175091652, −2.46537725512745291617344393151, −2.14653745759805693878357065749, −1.09869403552097888024626087514, 1.09869403552097888024626087514, 2.14653745759805693878357065749, 2.46537725512745291617344393151, 3.60830230138185988586175091652, 3.82699059385246898579623431730, 4.97263177724634349634715496710, 5.79058635469948905459477007171, 6.51643178541469402322785891964, 7.57130856174502142911339740771, 7.80485806116074002038213925499

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