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  − 1.77·3-s − 0.716·5-s + 0.155·9-s + 1.69·11-s − 4.41·13-s + 1.27·15-s + 1.35·17-s + 6.61·19-s − 4.94·23-s − 4.48·25-s + 5.05·27-s − 0.710·29-s + 3.41·31-s − 3.01·33-s − 5.21·37-s + 7.84·39-s − 41-s − 6.26·43-s − 0.111·45-s + 7.29·47-s − 2.41·51-s − 12.1·53-s − 1.21·55-s − 11.7·57-s − 1.69·59-s + 7.76·61-s + 3.16·65-s + ⋯
L(s)  = 1  − 1.02·3-s − 0.320·5-s + 0.0518·9-s + 0.511·11-s − 1.22·13-s + 0.328·15-s + 0.329·17-s + 1.51·19-s − 1.03·23-s − 0.897·25-s + 0.972·27-s − 0.132·29-s + 0.613·31-s − 0.524·33-s − 0.857·37-s + 1.25·39-s − 0.156·41-s − 0.955·43-s − 0.0166·45-s + 1.06·47-s − 0.337·51-s − 1.67·53-s − 0.164·55-s − 1.55·57-s − 0.220·59-s + 0.994·61-s + 0.392·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7770935169$
$L(\frac12)$  $\approx$  $0.7770935169$
$L(\frac{3}{2})$   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 \( 1 + 1.77T + 3T^{2} \)
5 \( 1 + 0.716T + 5T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
13 \( 1 + 4.41T + 13T^{2} \)
17 \( 1 - 1.35T + 17T^{2} \)
19 \( 1 - 6.61T + 19T^{2} \)
23 \( 1 + 4.94T + 23T^{2} \)
29 \( 1 + 0.710T + 29T^{2} \)
31 \( 1 - 3.41T + 31T^{2} \)
37 \( 1 + 5.21T + 37T^{2} \)
43 \( 1 + 6.26T + 43T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 1.69T + 59T^{2} \)
61 \( 1 - 7.76T + 61T^{2} \)
67 \( 1 - 3.13T + 67T^{2} \)
71 \( 1 - 3.54T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 8.90T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 + 6.08T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−7.82420780953476813831515298285, −7.00479951359984682145767260863, −6.49524098459498868294902924007, −5.54449119051509758132827385046, −5.27227911717776737359047529905, −4.40287351031428937742156067853, −3.59010355443660057502824670648, −2.71336930056397601560812243284, −1.59525522383135092139291866170, −0.46465366664918793496032799846, 0.46465366664918793496032799846, 1.59525522383135092139291866170, 2.71336930056397601560812243284, 3.59010355443660057502824670648, 4.40287351031428937742156067853, 5.27227911717776737359047529905, 5.54449119051509758132827385046, 6.49524098459498868294902924007, 7.00479951359984682145767260863, 7.82420780953476813831515298285

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