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  − 1.89·2-s + 3-s + 1.57·4-s + 1.57·5-s − 1.89·6-s + 0.800·8-s + 9-s − 2.98·10-s + 1.66·11-s + 1.57·12-s + 5.04·13-s + 1.57·15-s − 4.66·16-s − 6.35·17-s − 1.89·18-s − 0.691·19-s + 2.48·20-s − 3.15·22-s + 8.82·23-s + 0.800·24-s − 2.51·25-s − 9.54·26-s + 27-s − 6.64·29-s − 2.98·30-s − 2.35·31-s + 7.22·32-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.577·3-s + 0.788·4-s + 0.705·5-s − 0.772·6-s + 0.283·8-s + 0.333·9-s − 0.942·10-s + 0.502·11-s + 0.455·12-s + 1.39·13-s + 0.407·15-s − 1.16·16-s − 1.54·17-s − 0.445·18-s − 0.158·19-s + 0.555·20-s − 0.672·22-s + 1.84·23-s + 0.163·24-s − 0.502·25-s − 1.87·26-s + 0.192·27-s − 1.23·29-s − 0.544·30-s − 0.423·31-s + 1.27·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.519713738$
$L(\frac12)$  $\approx$  $1.519713738$
$L(\frac{3}{2})$   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 \( 1 + 1.89T + 2T^{2} \)
5 \( 1 - 1.57T + 5T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 - 5.04T + 13T^{2} \)
17 \( 1 + 6.35T + 17T^{2} \)
19 \( 1 + 0.691T + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
29 \( 1 + 6.64T + 29T^{2} \)
31 \( 1 + 2.35T + 31T^{2} \)
37 \( 1 - 9.02T + 37T^{2} \)
43 \( 1 + 9.96T + 43T^{2} \)
47 \( 1 - 1.33T + 47T^{2} \)
53 \( 1 - 1.66T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 0.870T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 - 2.79T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 2.62T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 3.21T + 97T^{2} \)
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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

−8.370830799494059086717251358891, −7.53705390074819106309444506936, −6.79402654142923043069727024570, −6.31953534434513971524197563743, −5.28373734483245046359074436086, −4.30784490054688919712410742117, −3.55183377681472187776939238284, −2.35233810266798006277017958435, −1.72342363493978399200393971813, −0.808217620795406463893468902869, 0.808217620795406463893468902869, 1.72342363493978399200393971813, 2.35233810266798006277017958435, 3.55183377681472187776939238284, 4.30784490054688919712410742117, 5.28373734483245046359074436086, 6.31953534434513971524197563743, 6.79402654142923043069727024570, 7.53705390074819106309444506936, 8.370830799494059086717251358891

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