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  + 1.50·2-s − 3-s + 0.272·4-s − 1.10·5-s − 1.50·6-s − 2.60·8-s + 9-s − 1.67·10-s + 3.74·11-s − 0.272·12-s − 4.04·13-s + 1.10·15-s − 4.47·16-s + 1.28·17-s + 1.50·18-s − 2.77·19-s − 0.302·20-s + 5.63·22-s + 8.14·23-s + 2.60·24-s − 3.77·25-s − 6.10·26-s − 27-s − 1.13·29-s + 1.67·30-s − 3.06·31-s − 1.53·32-s + ⋯
L(s)  = 1  + 1.06·2-s − 0.577·3-s + 0.136·4-s − 0.495·5-s − 0.615·6-s − 0.920·8-s + 0.333·9-s − 0.528·10-s + 1.12·11-s − 0.0786·12-s − 1.12·13-s + 0.286·15-s − 1.11·16-s + 0.311·17-s + 0.355·18-s − 0.636·19-s − 0.0675·20-s + 1.20·22-s + 1.69·23-s + 0.531·24-s − 0.754·25-s − 1.19·26-s − 0.192·27-s − 0.210·29-s + 0.305·30-s − 0.550·31-s − 0.270·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.753847517$
$L(\frac12)$  $\approx$  $1.753847517$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 1.50T + 2T^{2} \)
5 \( 1 + 1.10T + 5T^{2} \)
11 \( 1 - 3.74T + 11T^{2} \)
13 \( 1 + 4.04T + 13T^{2} \)
17 \( 1 - 1.28T + 17T^{2} \)
19 \( 1 + 2.77T + 19T^{2} \)
23 \( 1 - 8.14T + 23T^{2} \)
29 \( 1 + 1.13T + 29T^{2} \)
31 \( 1 + 3.06T + 31T^{2} \)
37 \( 1 + 8.99T + 37T^{2} \)
43 \( 1 - 5.07T + 43T^{2} \)
47 \( 1 - 1.77T + 47T^{2} \)
53 \( 1 + 1.43T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 6.29T + 61T^{2} \)
67 \( 1 - 1.53T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 7.32T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 0.862T + 89T^{2} \)
97 \( 1 - 2.95T + 97T^{2} \)
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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

−7.911740932917318918695820541553, −7.01980060725886525740277511697, −6.66300842931902430260290685450, −5.69414616589675751046133791704, −5.16929760710498993562827822130, −4.44544625116049382522883428710, −3.85447198813114642430450202693, −3.12427460888625254503019478308, −1.99587579605124676754065569873, −0.59816492981528670534481558681, 0.59816492981528670534481558681, 1.99587579605124676754065569873, 3.12427460888625254503019478308, 3.85447198813114642430450202693, 4.44544625116049382522883428710, 5.16929760710498993562827822130, 5.69414616589675751046133791704, 6.66300842931902430260290685450, 7.01980060725886525740277511697, 7.911740932917318918695820541553

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