Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.69·2-s + 3-s + 5.23·4-s − 3.40·5-s + 2.69·6-s + 8.70·8-s + 9-s − 9.16·10-s − 6.16·11-s + 5.23·12-s − 3.11·13-s − 3.40·15-s + 12.9·16-s − 2.71·17-s + 2.69·18-s − 7.15·19-s − 17.8·20-s − 16.5·22-s + 3.55·23-s + 8.70·24-s + 6.60·25-s − 8.38·26-s + 27-s − 1.79·29-s − 9.16·30-s + 6.90·31-s + 17.4·32-s + ⋯
L(s)  = 1  + 1.90·2-s + 0.577·3-s + 2.61·4-s − 1.52·5-s + 1.09·6-s + 3.07·8-s + 0.333·9-s − 2.89·10-s − 1.85·11-s + 1.51·12-s − 0.864·13-s − 0.879·15-s + 3.23·16-s − 0.659·17-s + 0.634·18-s − 1.64·19-s − 3.98·20-s − 3.53·22-s + 0.740·23-s + 1.77·24-s + 1.32·25-s − 1.64·26-s + 0.192·27-s − 0.332·29-s − 1.67·30-s + 1.24·31-s + 3.08·32-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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 2.69T + 2T^{2} \)
5 \( 1 + 3.40T + 5T^{2} \)
11 \( 1 + 6.16T + 11T^{2} \)
13 \( 1 + 3.11T + 13T^{2} \)
17 \( 1 + 2.71T + 17T^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
23 \( 1 - 3.55T + 23T^{2} \)
29 \( 1 + 1.79T + 29T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 + 8.76T + 37T^{2} \)
43 \( 1 - 3.02T + 43T^{2} \)
47 \( 1 - 4.99T + 47T^{2} \)
53 \( 1 + 2.58T + 53T^{2} \)
59 \( 1 + 9.63T + 59T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + 0.984T + 73T^{2} \)
79 \( 1 + 8.68T + 79T^{2} \)
83 \( 1 + 1.55T + 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 - 4.47T + 97T^{2} \)
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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

−7.40304144563853041241335078539, −7.13953974801104575102817339157, −6.19499663639082166110044028258, −5.24998970740744125011025491034, −4.47573206178076338324665017867, −4.36166768238589274296928613984, −3.21796474000599316486862676816, −2.79619802695387514961603007277, −2.01419209629349933652639767899, 0, 2.01419209629349933652639767899, 2.79619802695387514961603007277, 3.21796474000599316486862676816, 4.36166768238589274296928613984, 4.47573206178076338324665017867, 5.24998970740744125011025491034, 6.19499663639082166110044028258, 7.13953974801104575102817339157, 7.40304144563853041241335078539

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