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  − 0.631·2-s + 3-s − 1.60·4-s − 1.65·5-s − 0.631·6-s + 2.27·8-s + 9-s + 1.04·10-s + 2.82·11-s − 1.60·12-s − 3.15·13-s − 1.65·15-s + 1.76·16-s + 1.11·17-s − 0.631·18-s + 3.11·19-s + 2.65·20-s − 1.78·22-s − 6.51·23-s + 2.27·24-s − 2.24·25-s + 1.99·26-s + 27-s + 9.16·29-s + 1.04·30-s − 4.35·31-s − 5.66·32-s + ⋯
L(s)  = 1  − 0.446·2-s + 0.577·3-s − 0.800·4-s − 0.742·5-s − 0.257·6-s + 0.803·8-s + 0.333·9-s + 0.331·10-s + 0.850·11-s − 0.462·12-s − 0.876·13-s − 0.428·15-s + 0.442·16-s + 0.271·17-s − 0.148·18-s + 0.713·19-s + 0.594·20-s − 0.379·22-s − 1.35·23-s + 0.463·24-s − 0.449·25-s + 0.391·26-s + 0.192·27-s + 1.70·29-s + 0.191·30-s − 0.781·31-s − 1.00·32-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 \( 1 + 0.631T + 2T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 3.15T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 - 3.11T + 19T^{2} \)
23 \( 1 + 6.51T + 23T^{2} \)
29 \( 1 - 9.16T + 29T^{2} \)
31 \( 1 + 4.35T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
43 \( 1 + 0.684T + 43T^{2} \)
47 \( 1 - 6.17T + 47T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 - 7.71T + 59T^{2} \)
61 \( 1 - 0.419T + 61T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 - 7.39T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 0.195T + 79T^{2} \)
83 \( 1 + 2.50T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−7.900554933995541752695715902283, −7.28960025193714649524238589635, −6.52893995126055932771284011528, −5.39129292685125465767692033543, −4.72010214367152190658732943996, −3.87877484090846537980316560340, −3.49974531873276315667048797467, −2.23757775080955695155999057091, −1.17774125785693254246332959793, 0, 1.17774125785693254246332959793, 2.23757775080955695155999057091, 3.49974531873276315667048797467, 3.87877484090846537980316560340, 4.72010214367152190658732943996, 5.39129292685125465767692033543, 6.52893995126055932771284011528, 7.28960025193714649524238589635, 7.900554933995541752695715902283

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