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  + 0.470·2-s + 3-s − 1.77·4-s − 4.24·5-s + 0.470·6-s − 1.77·8-s + 9-s − 2·10-s − 1.47·11-s − 1.77·12-s + 0.249·13-s − 4.24·15-s + 2.71·16-s + 5.02·17-s + 0.470·18-s + 2.24·19-s + 7.55·20-s − 0.692·22-s − 6.24·23-s − 1.77·24-s + 13.0·25-s + 0.117·26-s + 27-s − 2.41·29-s − 2·30-s + 2.89·31-s + 4.83·32-s + ⋯
L(s)  = 1  + 0.332·2-s + 0.577·3-s − 0.889·4-s − 1.90·5-s + 0.192·6-s − 0.628·8-s + 0.333·9-s − 0.632·10-s − 0.443·11-s − 0.513·12-s + 0.0690·13-s − 1.09·15-s + 0.679·16-s + 1.21·17-s + 0.110·18-s + 0.515·19-s + 1.68·20-s − 0.147·22-s − 1.30·23-s − 0.363·24-s + 2.61·25-s + 0.0229·26-s + 0.192·27-s − 0.447·29-s − 0.365·30-s + 0.520·31-s + 0.855·32-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 - 0.470T + 2T^{2} \)
5 \( 1 + 4.24T + 5T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 - 0.249T + 13T^{2} \)
17 \( 1 - 5.02T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + 6.24T + 23T^{2} \)
29 \( 1 + 2.41T + 29T^{2} \)
31 \( 1 - 2.89T + 31T^{2} \)
37 \( 1 - 9.71T + 37T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 4.08T + 47T^{2} \)
53 \( 1 + 1.43T + 53T^{2} \)
59 \( 1 + 2.61T + 59T^{2} \)
61 \( 1 - 5.71T + 61T^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + 9.39T + 73T^{2} \)
79 \( 1 + 0.560T + 79T^{2} \)
83 \( 1 + 3.80T + 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 - 7.19T + 97T^{2} \)
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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

−7.78676204100997859939588684438, −7.42316092050796779346889251448, −6.27911594932678334296821837468, −5.32721283559812962615037161175, −4.62849428651522142898956675643, −3.91994287805849195431874347760, −3.49238255870543996453902242053, −2.74430060367268899706109205095, −1.07130706863150774851147310589, 0, 1.07130706863150774851147310589, 2.74430060367268899706109205095, 3.49238255870543996453902242053, 3.91994287805849195431874347760, 4.62849428651522142898956675643, 5.32721283559812962615037161175, 6.27911594932678334296821837468, 7.42316092050796779346889251448, 7.78676204100997859939588684438

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