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  + 0.154·2-s − 3-s − 1.97·4-s − 1.29·5-s − 0.154·6-s − 0.612·8-s + 9-s − 0.198·10-s − 3.77·11-s + 1.97·12-s + 5.50·13-s + 1.29·15-s + 3.85·16-s + 0.121·17-s + 0.154·18-s − 6.50·19-s + 2.54·20-s − 0.582·22-s + 2.09·23-s + 0.612·24-s − 3.33·25-s + 0.848·26-s − 27-s − 1.13·29-s + 0.198·30-s − 0.113·31-s + 1.82·32-s + ⋯
L(s)  = 1  + 0.108·2-s − 0.577·3-s − 0.988·4-s − 0.576·5-s − 0.0629·6-s − 0.216·8-s + 0.333·9-s − 0.0628·10-s − 1.13·11-s + 0.570·12-s + 1.52·13-s + 0.333·15-s + 0.964·16-s + 0.0294·17-s + 0.0363·18-s − 1.49·19-s + 0.570·20-s − 0.124·22-s + 0.436·23-s + 0.125·24-s − 0.667·25-s + 0.166·26-s − 0.192·27-s − 0.210·29-s + 0.0363·30-s − 0.0203·31-s + 0.321·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5814181165$
$L(\frac12)$  $\approx$  $0.5814181165$
$L(\frac{3}{2})$   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 \( 1 - 0.154T + 2T^{2} \)
5 \( 1 + 1.29T + 5T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
13 \( 1 - 5.50T + 13T^{2} \)
17 \( 1 - 0.121T + 17T^{2} \)
19 \( 1 + 6.50T + 19T^{2} \)
23 \( 1 - 2.09T + 23T^{2} \)
29 \( 1 + 1.13T + 29T^{2} \)
31 \( 1 + 0.113T + 31T^{2} \)
37 \( 1 + 4.51T + 37T^{2} \)
43 \( 1 - 4.61T + 43T^{2} \)
47 \( 1 + 3.21T + 47T^{2} \)
53 \( 1 + 3.27T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 2.67T + 61T^{2} \)
67 \( 1 - 7.34T + 67T^{2} \)
71 \( 1 - 3.10T + 71T^{2} \)
73 \( 1 + 0.731T + 73T^{2} \)
79 \( 1 + 2.88T + 79T^{2} \)
83 \( 1 + 6.55T + 83T^{2} \)
89 \( 1 + 3.36T + 89T^{2} \)
97 \( 1 + 6.52T + 97T^{2} \)
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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

−8.169050154478532830018642195346, −7.51630559579851989540009379295, −6.47920659886449079560530258361, −5.86541756765435921125613739917, −5.19375839416843497795064448712, −4.39024605191831554010800462795, −3.87525181426656814248457012272, −3.01278866553128700985793167117, −1.65002457261632633853397305202, −0.41319858891938600148978072086, 0.41319858891938600148978072086, 1.65002457261632633853397305202, 3.01278866553128700985793167117, 3.87525181426656814248457012272, 4.39024605191831554010800462795, 5.19375839416843497795064448712, 5.86541756765435921125613739917, 6.47920659886449079560530258361, 7.51630559579851989540009379295, 8.169050154478532830018642195346

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