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  + 1.62·2-s + 3-s + 0.644·4-s + 1.67·5-s + 1.62·6-s − 2.20·8-s + 9-s + 2.72·10-s + 2.72·11-s + 0.644·12-s − 2.59·13-s + 1.67·15-s − 4.87·16-s + 2.49·17-s + 1.62·18-s − 0.765·19-s + 1.07·20-s + 4.43·22-s + 1.55·23-s − 2.20·24-s − 2.19·25-s − 4.21·26-s + 27-s + 9.11·29-s + 2.72·30-s + 8.26·31-s − 3.51·32-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.577·3-s + 0.322·4-s + 0.748·5-s + 0.663·6-s − 0.779·8-s + 0.333·9-s + 0.860·10-s + 0.821·11-s + 0.186·12-s − 0.718·13-s + 0.432·15-s − 1.21·16-s + 0.605·17-s + 0.383·18-s − 0.175·19-s + 0.241·20-s + 0.944·22-s + 0.323·23-s − 0.449·24-s − 0.439·25-s − 0.826·26-s + 0.192·27-s + 1.69·29-s + 0.496·30-s + 1.48·31-s − 0.621·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.097285518$
$L(\frac12)$  $\approx$  $5.097285518$
$L(\frac{3}{2})$   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 \( 1 - 1.62T + 2T^{2} \)
5 \( 1 - 1.67T + 5T^{2} \)
11 \( 1 - 2.72T + 11T^{2} \)
13 \( 1 + 2.59T + 13T^{2} \)
17 \( 1 - 2.49T + 17T^{2} \)
19 \( 1 + 0.765T + 19T^{2} \)
23 \( 1 - 1.55T + 23T^{2} \)
29 \( 1 - 9.11T + 29T^{2} \)
31 \( 1 - 8.26T + 31T^{2} \)
37 \( 1 + 3.21T + 37T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 5.81T + 47T^{2} \)
53 \( 1 + 4.59T + 53T^{2} \)
59 \( 1 + 4.77T + 59T^{2} \)
61 \( 1 + 0.867T + 61T^{2} \)
67 \( 1 - 3.89T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 2.23T + 73T^{2} \)
79 \( 1 - 8.10T + 79T^{2} \)
83 \( 1 - 8.58T + 83T^{2} \)
89 \( 1 - 1.25T + 89T^{2} \)
97 \( 1 - 8.55T + 97T^{2} \)
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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

−8.056394254929366769252105732127, −7.21529169690205461757379178375, −6.34925252594322355516343124740, −5.99718980121794281601192869205, −5.00000437322537542679478361456, −4.50283409876726704520642859282, −3.70236307519436342687385386726, −2.86479688708274493416816186122, −2.26402994211093032574895750899, −1.00741684846325318275026327863, 1.00741684846325318275026327863, 2.26402994211093032574895750899, 2.86479688708274493416816186122, 3.70236307519436342687385386726, 4.50283409876726704520642859282, 5.00000437322537542679478361456, 5.99718980121794281601192869205, 6.34925252594322355516343124740, 7.21529169690205461757379178375, 8.056394254929366769252105732127

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