Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.87·2-s + 3-s + 1.51·4-s − 1.33·5-s + 1.87·6-s − 0.902·8-s + 9-s − 2.49·10-s + 1.19·11-s + 1.51·12-s + 0.956·13-s − 1.33·15-s − 4.73·16-s + 3.26·17-s + 1.87·18-s − 6.16·19-s − 2.02·20-s + 2.23·22-s − 6.99·23-s − 0.902·24-s − 3.22·25-s + 1.79·26-s + 27-s − 2.28·29-s − 2.49·30-s − 1.84·31-s − 7.06·32-s + ⋯
L(s)  = 1  + 1.32·2-s + 0.577·3-s + 0.759·4-s − 0.595·5-s + 0.765·6-s − 0.318·8-s + 0.333·9-s − 0.789·10-s + 0.359·11-s + 0.438·12-s + 0.265·13-s − 0.343·15-s − 1.18·16-s + 0.791·17-s + 0.442·18-s − 1.41·19-s − 0.452·20-s + 0.476·22-s − 1.45·23-s − 0.184·24-s − 0.645·25-s + 0.351·26-s + 0.192·27-s − 0.424·29-s − 0.455·30-s − 0.331·31-s − 1.24·32-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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 1.87T + 2T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 - 0.956T + 13T^{2} \)
17 \( 1 - 3.26T + 17T^{2} \)
19 \( 1 + 6.16T + 19T^{2} \)
23 \( 1 + 6.99T + 23T^{2} \)
29 \( 1 + 2.28T + 29T^{2} \)
31 \( 1 + 1.84T + 31T^{2} \)
37 \( 1 + 6.15T + 37T^{2} \)
43 \( 1 + 5.98T + 43T^{2} \)
47 \( 1 - 6.24T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 4.26T + 59T^{2} \)
61 \( 1 - 4.85T + 61T^{2} \)
67 \( 1 - 3.76T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 6.06T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 2.10T + 83T^{2} \)
89 \( 1 + 9.71T + 89T^{2} \)
97 \( 1 - 0.622T + 97T^{2} \)
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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

−7.60421972685412005366042038361, −6.93777361951846633059664886595, −6.07293209687046148827799087246, −5.58893665359499562009787693826, −4.56277356366568181878386972857, −3.89803933464193831590958349693, −3.64378831461821399549200942364, −2.59256329527972157761309898118, −1.73808363524645536915559535152, 0, 1.73808363524645536915559535152, 2.59256329527972157761309898118, 3.64378831461821399549200942364, 3.89803933464193831590958349693, 4.56277356366568181878386972857, 5.58893665359499562009787693826, 6.07293209687046148827799087246, 6.93777361951846633059664886595, 7.60421972685412005366042038361

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