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  − 1.96·2-s − 3-s + 1.87·4-s + 4.10·5-s + 1.96·6-s + 0.241·8-s + 9-s − 8.09·10-s + 4.87·11-s − 1.87·12-s + 2.68·13-s − 4.10·15-s − 4.23·16-s + 3.34·17-s − 1.96·18-s + 0.186·19-s + 7.71·20-s − 9.60·22-s + 5.74·23-s − 0.241·24-s + 11.8·25-s − 5.29·26-s − 27-s + 2.28·29-s + 8.09·30-s − 0.783·31-s + 7.84·32-s + ⋯
L(s)  = 1  − 1.39·2-s − 0.577·3-s + 0.938·4-s + 1.83·5-s + 0.803·6-s + 0.0854·8-s + 0.333·9-s − 2.55·10-s + 1.47·11-s − 0.541·12-s + 0.745·13-s − 1.06·15-s − 1.05·16-s + 0.811·17-s − 0.464·18-s + 0.0426·19-s + 1.72·20-s − 2.04·22-s + 1.19·23-s − 0.0493·24-s + 2.37·25-s − 1.03·26-s − 0.192·27-s + 0.425·29-s + 1.47·30-s − 0.140·31-s + 1.38·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.575715783$
$L(\frac12)$  $\approx$  $1.575715783$
$L(\frac{3}{2})$   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 \( 1 + 1.96T + 2T^{2} \)
5 \( 1 - 4.10T + 5T^{2} \)
11 \( 1 - 4.87T + 11T^{2} \)
13 \( 1 - 2.68T + 13T^{2} \)
17 \( 1 - 3.34T + 17T^{2} \)
19 \( 1 - 0.186T + 19T^{2} \)
23 \( 1 - 5.74T + 23T^{2} \)
29 \( 1 - 2.28T + 29T^{2} \)
31 \( 1 + 0.783T + 31T^{2} \)
37 \( 1 - 1.26T + 37T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + 2.29T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 1.36T + 61T^{2} \)
67 \( 1 + 2.74T + 67T^{2} \)
71 \( 1 - 6.59T + 71T^{2} \)
73 \( 1 + 3.81T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 3.54T + 83T^{2} \)
89 \( 1 + 9.08T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−8.485257278696598302449306269458, −7.19970717004787856814216097848, −6.75261526308314480412230046897, −6.14266232836332169866933108652, −5.45849781868906407221630958406, −4.65661443326395690141412368577, −3.48629314302270333611807659450, −2.28803937645491997692044568999, −1.34468878980685697487639596019, −1.03993478179515962516238115299, 1.03993478179515962516238115299, 1.34468878980685697487639596019, 2.28803937645491997692044568999, 3.48629314302270333611807659450, 4.65661443326395690141412368577, 5.45849781868906407221630958406, 6.14266232836332169866933108652, 6.75261526308314480412230046897, 7.19970717004787856814216097848, 8.485257278696598302449306269458

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