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  − 0.423·2-s − 3-s − 1.82·4-s − 3.41·5-s + 0.423·6-s + 1.61·8-s + 9-s + 1.44·10-s + 1.20·11-s + 1.82·12-s − 1.26·13-s + 3.41·15-s + 2.95·16-s − 2.17·17-s − 0.423·18-s + 7.30·19-s + 6.21·20-s − 0.512·22-s + 4.90·23-s − 1.61·24-s + 6.67·25-s + 0.535·26-s − 27-s + 0.506·29-s − 1.44·30-s − 6.04·31-s − 4.49·32-s + ⋯
L(s)  = 1  − 0.299·2-s − 0.577·3-s − 0.910·4-s − 1.52·5-s + 0.173·6-s + 0.572·8-s + 0.333·9-s + 0.457·10-s + 0.364·11-s + 0.525·12-s − 0.350·13-s + 0.882·15-s + 0.738·16-s − 0.527·17-s − 0.0998·18-s + 1.67·19-s + 1.39·20-s − 0.109·22-s + 1.02·23-s − 0.330·24-s + 1.33·25-s + 0.104·26-s − 0.192·27-s + 0.0940·29-s − 0.264·30-s − 1.08·31-s − 0.793·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5190875140$
$L(\frac12)$  $\approx$  $0.5190875140$
$L(\frac{3}{2})$   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 \( 1 + 0.423T + 2T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 - 1.20T + 11T^{2} \)
13 \( 1 + 1.26T + 13T^{2} \)
17 \( 1 + 2.17T + 17T^{2} \)
19 \( 1 - 7.30T + 19T^{2} \)
23 \( 1 - 4.90T + 23T^{2} \)
29 \( 1 - 0.506T + 29T^{2} \)
31 \( 1 + 6.04T + 31T^{2} \)
37 \( 1 - 6.73T + 37T^{2} \)
43 \( 1 + 7.21T + 43T^{2} \)
47 \( 1 + 5.13T + 47T^{2} \)
53 \( 1 + 7.67T + 53T^{2} \)
59 \( 1 + 3.05T + 59T^{2} \)
61 \( 1 + 1.50T + 61T^{2} \)
67 \( 1 - 2.66T + 67T^{2} \)
71 \( 1 + 8.67T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 7.94T + 79T^{2} \)
83 \( 1 - 5.53T + 83T^{2} \)
89 \( 1 + 4.64T + 89T^{2} \)
97 \( 1 - 1.13T + 97T^{2} \)
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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

−7.969361133693377788794522752594, −7.47175989297729098928903257775, −6.93278010462156789641217910898, −5.85730287796066189621579298570, −4.91147096142609081818713854672, −4.61597045561202823570336268783, −3.69970572364310779390164430697, −3.11996846999201960818534974457, −1.39120439228513815194501576740, −0.45327296601441005232904729583, 0.45327296601441005232904729583, 1.39120439228513815194501576740, 3.11996846999201960818534974457, 3.69970572364310779390164430697, 4.61597045561202823570336268783, 4.91147096142609081818713854672, 5.85730287796066189621579298570, 6.93278010462156789641217910898, 7.47175989297729098928903257775, 7.969361133693377788794522752594

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