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  + 2.74·2-s + 3-s + 5.51·4-s + 0.315·5-s + 2.74·6-s + 9.64·8-s + 9-s + 0.866·10-s − 3.45·11-s + 5.51·12-s + 0.303·13-s + 0.315·15-s + 15.4·16-s − 5.96·17-s + 2.74·18-s + 8.54·19-s + 1.74·20-s − 9.47·22-s + 4.88·23-s + 9.64·24-s − 4.90·25-s + 0.831·26-s + 27-s + 10.1·29-s + 0.866·30-s − 6.98·31-s + 22.9·32-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.75·4-s + 0.141·5-s + 1.11·6-s + 3.41·8-s + 0.333·9-s + 0.273·10-s − 1.04·11-s + 1.59·12-s + 0.0841·13-s + 0.0815·15-s + 3.85·16-s − 1.44·17-s + 0.646·18-s + 1.96·19-s + 0.389·20-s − 2.01·22-s + 1.01·23-s + 1.96·24-s − 0.980·25-s + 0.163·26-s + 0.192·27-s + 1.88·29-s + 0.158·30-s − 1.25·31-s + 4.06·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.742137188$
$L(\frac12)$  $\approx$  $9.742137188$
$L(\frac{3}{2})$   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 \( 1 - 2.74T + 2T^{2} \)
5 \( 1 - 0.315T + 5T^{2} \)
11 \( 1 + 3.45T + 11T^{2} \)
13 \( 1 - 0.303T + 13T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 - 8.54T + 19T^{2} \)
23 \( 1 - 4.88T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 6.98T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
43 \( 1 - 0.387T + 43T^{2} \)
47 \( 1 - 4.04T + 47T^{2} \)
53 \( 1 - 3.33T + 53T^{2} \)
59 \( 1 + 7.81T + 59T^{2} \)
61 \( 1 + 9.55T + 61T^{2} \)
67 \( 1 - 3.14T + 67T^{2} \)
71 \( 1 + 0.756T + 71T^{2} \)
73 \( 1 - 0.212T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 8.92T + 83T^{2} \)
89 \( 1 - 0.416T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.54400532156290061232081023989, −7.39839583044295120422813181184, −6.43424345815880249766168141849, −5.78521303733768358603121621408, −5.01508684812763160650882344766, −4.53970817195061802594095369858, −3.66665961196484400050526655846, −2.82525549090222440197453586630, −2.49147230481503767211463012558, −1.33130363526741436681324458158, 1.33130363526741436681324458158, 2.49147230481503767211463012558, 2.82525549090222440197453586630, 3.66665961196484400050526655846, 4.53970817195061802594095369858, 5.01508684812763160650882344766, 5.78521303733768358603121621408, 6.43424345815880249766168141849, 7.39839583044295120422813181184, 7.54400532156290061232081023989

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