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  + 1.76·2-s − 3-s + 1.10·4-s − 3.51·5-s − 1.76·6-s − 1.57·8-s + 9-s − 6.19·10-s − 0.153·11-s − 1.10·12-s + 2.35·13-s + 3.51·15-s − 4.98·16-s − 2.14·17-s + 1.76·18-s − 1.53·19-s − 3.89·20-s − 0.270·22-s − 3.30·23-s + 1.57·24-s + 7.34·25-s + 4.14·26-s − 27-s − 4.53·29-s + 6.19·30-s + 0.462·31-s − 5.64·32-s + ⋯
L(s)  = 1  + 1.24·2-s − 0.577·3-s + 0.554·4-s − 1.57·5-s − 0.719·6-s − 0.555·8-s + 0.333·9-s − 1.95·10-s − 0.0462·11-s − 0.319·12-s + 0.652·13-s + 0.907·15-s − 1.24·16-s − 0.520·17-s + 0.415·18-s − 0.353·19-s − 0.870·20-s − 0.0576·22-s − 0.689·23-s + 0.320·24-s + 1.46·25-s + 0.813·26-s − 0.192·27-s − 0.841·29-s + 1.13·30-s + 0.0830·31-s − 0.998·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.189992361$
$L(\frac12)$  $\approx$  $1.189992361$
$L(\frac{3}{2})$   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 \( 1 - 1.76T + 2T^{2} \)
5 \( 1 + 3.51T + 5T^{2} \)
11 \( 1 + 0.153T + 11T^{2} \)
13 \( 1 - 2.35T + 13T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 + 3.30T + 23T^{2} \)
29 \( 1 + 4.53T + 29T^{2} \)
31 \( 1 - 0.462T + 31T^{2} \)
37 \( 1 + 7.31T + 37T^{2} \)
43 \( 1 - 3.68T + 43T^{2} \)
47 \( 1 - 0.200T + 47T^{2} \)
53 \( 1 + 8.09T + 53T^{2} \)
59 \( 1 + 0.519T + 59T^{2} \)
61 \( 1 + 2.10T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 5.11T + 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 + 2.88T + 79T^{2} \)
83 \( 1 - 0.349T + 83T^{2} \)
89 \( 1 - 1.87T + 89T^{2} \)
97 \( 1 + 9.30T + 97T^{2} \)
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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

−7.969093679125038769380926686963, −7.16452682030930625113907481189, −6.51042853105840784400739492324, −5.84047155101965699973831622649, −5.00742293135935682811369331440, −4.41722277115228448721127788688, −3.76599130895314934051712405594, −3.33317781049080442061146920682, −2.05756731137298783008072924090, −0.46886837091238763160572202084, 0.46886837091238763160572202084, 2.05756731137298783008072924090, 3.33317781049080442061146920682, 3.76599130895314934051712405594, 4.41722277115228448721127788688, 5.00742293135935682811369331440, 5.84047155101965699973831622649, 6.51042853105840784400739492324, 7.16452682030930625113907481189, 7.969093679125038769380926686963

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