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  − 1.51·2-s + 3-s + 0.291·4-s − 3.80·5-s − 1.51·6-s + 2.58·8-s + 9-s + 5.76·10-s + 1.15·11-s + 0.291·12-s + 4.56·13-s − 3.80·15-s − 4.49·16-s + 6.92·17-s − 1.51·18-s − 2.40·19-s − 1.11·20-s − 1.75·22-s + 3.28·23-s + 2.58·24-s + 9.47·25-s − 6.90·26-s + 27-s + 1.28·29-s + 5.76·30-s + 0.334·31-s + 1.63·32-s + ⋯
L(s)  = 1  − 1.07·2-s + 0.577·3-s + 0.145·4-s − 1.70·5-s − 0.618·6-s + 0.914·8-s + 0.333·9-s + 1.82·10-s + 0.348·11-s + 0.0842·12-s + 1.26·13-s − 0.982·15-s − 1.12·16-s + 1.67·17-s − 0.356·18-s − 0.551·19-s − 0.248·20-s − 0.373·22-s + 0.684·23-s + 0.527·24-s + 1.89·25-s − 1.35·26-s + 0.192·27-s + 0.237·29-s + 1.05·30-s + 0.0601·31-s + 0.289·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.054810800$
$L(\frac12)$  $\approx$  $1.054810800$
$L(\frac{3}{2})$   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 \( 1 + 1.51T + 2T^{2} \)
5 \( 1 + 3.80T + 5T^{2} \)
11 \( 1 - 1.15T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 + 2.40T + 19T^{2} \)
23 \( 1 - 3.28T + 23T^{2} \)
29 \( 1 - 1.28T + 29T^{2} \)
31 \( 1 - 0.334T + 31T^{2} \)
37 \( 1 - 1.13T + 37T^{2} \)
43 \( 1 - 2.12T + 43T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + 2.25T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 2.57T + 61T^{2} \)
67 \( 1 + 6.21T + 67T^{2} \)
71 \( 1 - 2.30T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 2.58T + 83T^{2} \)
89 \( 1 - 9.21T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−8.110018200936449556583158731680, −7.62550289804334851928003002546, −7.16597524928568455439486334737, −6.18163077848147230403467856515, −5.03602572938532498351463043348, −4.10768504461385237863519027302, −3.74432385229345803056199990238, −2.87126707878407130751757385625, −1.38766469162176339825603674524, −0.70645246722566756506017658707, 0.70645246722566756506017658707, 1.38766469162176339825603674524, 2.87126707878407130751757385625, 3.74432385229345803056199990238, 4.10768504461385237863519027302, 5.03602572938532498351463043348, 6.18163077848147230403467856515, 7.16597524928568455439486334737, 7.62550289804334851928003002546, 8.110018200936449556583158731680

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