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  + 1.06·3-s − 0.715·5-s − 1.86·9-s − 0.203·11-s − 4.07·13-s − 0.763·15-s + 1.49·17-s − 1.74·19-s − 4.09·23-s − 4.48·25-s − 5.18·27-s − 5.13·29-s − 3.89·31-s − 0.216·33-s + 9.19·37-s − 4.34·39-s − 41-s + 1.74·43-s + 1.33·45-s + 6.43·47-s + 1.59·51-s + 6.61·53-s + 0.145·55-s − 1.86·57-s + 12.8·59-s + 14.5·61-s + 2.91·65-s + ⋯
L(s)  = 1  + 0.615·3-s − 0.319·5-s − 0.620·9-s − 0.0612·11-s − 1.13·13-s − 0.197·15-s + 0.363·17-s − 0.400·19-s − 0.854·23-s − 0.897·25-s − 0.998·27-s − 0.953·29-s − 0.699·31-s − 0.0377·33-s + 1.51·37-s − 0.696·39-s − 0.156·41-s + 0.266·43-s + 0.198·45-s + 0.938·47-s + 0.223·51-s + 0.908·53-s + 0.0196·55-s − 0.246·57-s + 1.67·59-s + 1.85·61-s + 0.361·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.513232468$
$L(\frac12)$  $\approx$  $1.513232468$
$L(\frac{3}{2})$   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 \( 1 - 1.06T + 3T^{2} \)
5 \( 1 + 0.715T + 5T^{2} \)
11 \( 1 + 0.203T + 11T^{2} \)
13 \( 1 + 4.07T + 13T^{2} \)
17 \( 1 - 1.49T + 17T^{2} \)
19 \( 1 + 1.74T + 19T^{2} \)
23 \( 1 + 4.09T + 23T^{2} \)
29 \( 1 + 5.13T + 29T^{2} \)
31 \( 1 + 3.89T + 31T^{2} \)
37 \( 1 - 9.19T + 37T^{2} \)
43 \( 1 - 1.74T + 43T^{2} \)
47 \( 1 - 6.43T + 47T^{2} \)
53 \( 1 - 6.61T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 14.5T + 61T^{2} \)
67 \( 1 + 0.105T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 6.88T + 79T^{2} \)
83 \( 1 + 2.43T + 83T^{2} \)
89 \( 1 + 4.37T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−7.76757831034988734191946053654, −7.44103241580292112766046226329, −6.47931589183108057458256219595, −5.65282593299150694410314319720, −5.14370926187648510069233882154, −3.97126160142866225437000288699, −3.68697897424732854206218047969, −2.43319898561093060756990238707, −2.18603764386119010951183720362, −0.55481992855078839024431896340, 0.55481992855078839024431896340, 2.18603764386119010951183720362, 2.43319898561093060756990238707, 3.68697897424732854206218047969, 3.97126160142866225437000288699, 5.14370926187648510069233882154, 5.65282593299150694410314319720, 6.47931589183108057458256219595, 7.44103241580292112766046226329, 7.76757831034988734191946053654

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