Properties

Degree 2
Conductor $ 3 \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  + 2.55·2-s + 3-s + 4.52·4-s + 2.03·5-s + 2.55·6-s + 6.44·8-s + 9-s + 5.20·10-s + 2.19·11-s + 4.52·12-s + 4.05·13-s + 2.03·15-s + 7.41·16-s − 4.84·17-s + 2.55·18-s + 7.41·19-s + 9.22·20-s + 5.61·22-s − 4.70·23-s + 6.44·24-s − 0.839·25-s + 10.3·26-s + 27-s − 10.1·29-s + 5.20·30-s − 3.76·31-s + 6.04·32-s + ⋯
L(s)  = 1  + 1.80·2-s + 0.577·3-s + 2.26·4-s + 0.912·5-s + 1.04·6-s + 2.27·8-s + 0.333·9-s + 1.64·10-s + 0.662·11-s + 1.30·12-s + 1.12·13-s + 0.526·15-s + 1.85·16-s − 1.17·17-s + 0.602·18-s + 1.70·19-s + 2.06·20-s + 1.19·22-s − 0.980·23-s + 1.31·24-s − 0.167·25-s + 2.03·26-s + 0.192·27-s − 1.87·29-s + 0.951·30-s − 0.675·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $10.17794565$
$L(\frac12)$  $\approx$  $10.17794565$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 2.55T + 2T^{2} \)
5 \( 1 - 2.03T + 5T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 - 4.05T + 13T^{2} \)
17 \( 1 + 4.84T + 17T^{2} \)
19 \( 1 - 7.41T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 + 3.76T + 31T^{2} \)
37 \( 1 + 6.51T + 37T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 - 0.528T + 47T^{2} \)
53 \( 1 - 7.25T + 53T^{2} \)
59 \( 1 - 7.53T + 59T^{2} \)
61 \( 1 - 1.26T + 61T^{2} \)
67 \( 1 + 8.72T + 67T^{2} \)
71 \( 1 + 1.95T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + 2.93T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + 4.93T + 89T^{2} \)
97 \( 1 + 4.56T + 97T^{2} \)
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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

−7.74155256525781752367350396594, −7.11818479211960895787710246482, −6.42378182634637101442313591716, −5.65149234536993054711968312551, −5.44352531000484869275045668974, −4.10988981126856841993398390310, −3.89589340844800743794704531853, −2.99790518799137699470984861850, −2.08786049976310395982685198149, −1.51134496415935205468792478518, 1.51134496415935205468792478518, 2.08786049976310395982685198149, 2.99790518799137699470984861850, 3.89589340844800743794704531853, 4.10988981126856841993398390310, 5.44352531000484869275045668974, 5.65149234536993054711968312551, 6.42378182634637101442313591716, 7.11818479211960895787710246482, 7.74155256525781752367350396594

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