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  + 1.66·2-s + 3-s + 0.782·4-s + 4.32·5-s + 1.66·6-s − 2.03·8-s + 9-s + 7.21·10-s + 5.84·11-s + 0.782·12-s + 4.56·13-s + 4.32·15-s − 4.95·16-s + 1.48·17-s + 1.66·18-s − 7.25·19-s + 3.38·20-s + 9.75·22-s + 3.99·23-s − 2.03·24-s + 13.6·25-s + 7.60·26-s + 27-s − 3.99·29-s + 7.21·30-s − 3.33·31-s − 4.19·32-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.577·3-s + 0.391·4-s + 1.93·5-s + 0.680·6-s − 0.718·8-s + 0.333·9-s + 2.28·10-s + 1.76·11-s + 0.225·12-s + 1.26·13-s + 1.11·15-s − 1.23·16-s + 0.360·17-s + 0.393·18-s − 1.66·19-s + 0.756·20-s + 2.08·22-s + 0.832·23-s − 0.414·24-s + 2.73·25-s + 1.49·26-s + 0.192·27-s − 0.742·29-s + 1.31·30-s − 0.599·31-s − 0.742·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.184065130$
$L(\frac12)$  $\approx$  $7.184065130$
$L(\frac{3}{2})$   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 \( 1 - 1.66T + 2T^{2} \)
5 \( 1 - 4.32T + 5T^{2} \)
11 \( 1 - 5.84T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 1.48T + 17T^{2} \)
19 \( 1 + 7.25T + 19T^{2} \)
23 \( 1 - 3.99T + 23T^{2} \)
29 \( 1 + 3.99T + 29T^{2} \)
31 \( 1 + 3.33T + 31T^{2} \)
37 \( 1 - 0.681T + 37T^{2} \)
43 \( 1 + 7.66T + 43T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 + 2.70T + 53T^{2} \)
59 \( 1 - 0.715T + 59T^{2} \)
61 \( 1 - 3.33T + 61T^{2} \)
67 \( 1 + 7.99T + 67T^{2} \)
71 \( 1 - 7.51T + 71T^{2} \)
73 \( 1 + 0.453T + 73T^{2} \)
79 \( 1 + 6.22T + 79T^{2} \)
83 \( 1 + 7.75T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 7.48T + 97T^{2} \)
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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

−8.328941342506777299237022507695, −6.70860010846850625818754567512, −6.59556360372128840204687013465, −5.91192888279322133732520731719, −5.27200956625226922458654371612, −4.35878621443967075979020064664, −3.70311655176318088951502799959, −2.95487372908940787764055850383, −1.94076791486321292222352094319, −1.35347719470437705792573388618, 1.35347719470437705792573388618, 1.94076791486321292222352094319, 2.95487372908940787764055850383, 3.70311655176318088951502799959, 4.35878621443967075979020064664, 5.27200956625226922458654371612, 5.91192888279322133732520731719, 6.59556360372128840204687013465, 6.70860010846850625818754567512, 8.328941342506777299237022507695

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