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  + 2.90·3-s + 0.506·5-s + 5.44·9-s + 5.70·11-s + 2.51·13-s + 1.47·15-s + 5.03·17-s + 5.78·19-s + 1.51·23-s − 4.74·25-s + 7.11·27-s + 6.63·29-s − 3.27·31-s + 16.5·33-s + 9.82·37-s + 7.30·39-s − 41-s − 8.37·43-s + 2.75·45-s − 12.2·47-s + 14.6·51-s − 12.1·53-s + 2.88·55-s + 16.8·57-s − 8.79·59-s − 6.07·61-s + 1.27·65-s + ⋯
L(s)  = 1  + 1.67·3-s + 0.226·5-s + 1.81·9-s + 1.72·11-s + 0.697·13-s + 0.379·15-s + 1.22·17-s + 1.32·19-s + 0.316·23-s − 0.948·25-s + 1.36·27-s + 1.23·29-s − 0.588·31-s + 2.88·33-s + 1.61·37-s + 1.16·39-s − 0.156·41-s − 1.27·43-s + 0.410·45-s − 1.78·47-s + 2.04·51-s − 1.66·53-s + 0.389·55-s + 2.22·57-s − 1.14·59-s − 0.777·61-s + 0.157·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.477673052$
$L(\frac12)$  $\approx$  $5.477673052$
$L(\frac{3}{2})$   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 \( 1 - 2.90T + 3T^{2} \)
5 \( 1 - 0.506T + 5T^{2} \)
11 \( 1 - 5.70T + 11T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 - 5.03T + 17T^{2} \)
19 \( 1 - 5.78T + 19T^{2} \)
23 \( 1 - 1.51T + 23T^{2} \)
29 \( 1 - 6.63T + 29T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 - 9.82T + 37T^{2} \)
43 \( 1 + 8.37T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 8.79T + 59T^{2} \)
61 \( 1 + 6.07T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 - 4.85T + 73T^{2} \)
79 \( 1 + 2.19T + 79T^{2} \)
83 \( 1 - 0.906T + 83T^{2} \)
89 \( 1 + 5.03T + 89T^{2} \)
97 \( 1 + 5.21T + 97T^{2} \)
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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

−7.972133433491794801034316543333, −7.35127365283963501977496508765, −6.52445565177899844944649187069, −5.93167104647603265500485932019, −4.80862856538765537920629370290, −4.03010234511446700598199818048, −3.27911353937339477231591428898, −3.00956221426941375483071283047, −1.57989205777733924488774034894, −1.32503772785823084353900318680, 1.32503772785823084353900318680, 1.57989205777733924488774034894, 3.00956221426941375483071283047, 3.27911353937339477231591428898, 4.03010234511446700598199818048, 4.80862856538765537920629370290, 5.93167104647603265500485932019, 6.52445565177899844944649187069, 7.35127365283963501977496508765, 7.972133433491794801034316543333

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