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  + 2.11·3-s + 3.93·5-s + 1.48·9-s + 4.13·11-s + 1.64·13-s + 8.33·15-s − 5.20·17-s − 6.58·19-s + 7.17·23-s + 10.4·25-s − 3.20·27-s − 9.38·29-s + 6.46·31-s + 8.74·33-s + 11.3·37-s + 3.48·39-s − 41-s − 0.742·43-s + 5.84·45-s + 0.264·47-s − 11.0·51-s − 0.838·53-s + 16.2·55-s − 13.9·57-s − 5.28·59-s + 2.56·61-s + 6.47·65-s + ⋯
L(s)  = 1  + 1.22·3-s + 1.76·5-s + 0.494·9-s + 1.24·11-s + 0.456·13-s + 2.15·15-s − 1.26·17-s − 1.51·19-s + 1.49·23-s + 2.09·25-s − 0.617·27-s − 1.74·29-s + 1.16·31-s + 1.52·33-s + 1.86·37-s + 0.557·39-s − 0.156·41-s − 0.113·43-s + 0.871·45-s + 0.0385·47-s − 1.54·51-s − 0.115·53-s + 2.19·55-s − 1.84·57-s − 0.688·59-s + 0.328·61-s + 0.802·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.077318157$
$L(\frac12)$  $\approx$  $5.077318157$
$L(\frac{3}{2})$   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 \( 1 - 2.11T + 3T^{2} \)
5 \( 1 - 3.93T + 5T^{2} \)
11 \( 1 - 4.13T + 11T^{2} \)
13 \( 1 - 1.64T + 13T^{2} \)
17 \( 1 + 5.20T + 17T^{2} \)
19 \( 1 + 6.58T + 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 + 9.38T + 29T^{2} \)
31 \( 1 - 6.46T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
43 \( 1 + 0.742T + 43T^{2} \)
47 \( 1 - 0.264T + 47T^{2} \)
53 \( 1 + 0.838T + 53T^{2} \)
59 \( 1 + 5.28T + 59T^{2} \)
61 \( 1 - 2.56T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 5.81T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 - 2.51T + 79T^{2} \)
83 \( 1 + 8.61T + 83T^{2} \)
89 \( 1 - 1.18T + 89T^{2} \)
97 \( 1 - 6.32T + 97T^{2} \)
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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

−8.042018354085624567429060177014, −6.98309536301419965322422219825, −6.39892348905351694088828120413, −6.02413713691096983464293899580, −4.97056262997434465601680717065, −4.19215669731338100283166330954, −3.39121586197161735480121582333, −2.36383734985147367418783203087, −2.10937192480196111993075967523, −1.11903954385246266853464908222, 1.11903954385246266853464908222, 2.10937192480196111993075967523, 2.36383734985147367418783203087, 3.39121586197161735480121582333, 4.19215669731338100283166330954, 4.97056262997434465601680717065, 6.02413713691096983464293899580, 6.39892348905351694088828120413, 6.98309536301419965322422219825, 8.042018354085624567429060177014

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