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  − 0.308·2-s − 3-s − 1.90·4-s + 0.898·5-s + 0.308·6-s + 1.20·8-s + 9-s − 0.277·10-s + 1.38·11-s + 1.90·12-s − 6.64·13-s − 0.898·15-s + 3.43·16-s − 6.17·17-s − 0.308·18-s − 5.46·19-s − 1.71·20-s − 0.429·22-s + 4.76·23-s − 1.20·24-s − 4.19·25-s + 2.05·26-s − 27-s − 6.78·29-s + 0.277·30-s + 6.37·31-s − 3.47·32-s + ⋯
L(s)  = 1  − 0.218·2-s − 0.577·3-s − 0.952·4-s + 0.401·5-s + 0.126·6-s + 0.426·8-s + 0.333·9-s − 0.0877·10-s + 0.418·11-s + 0.549·12-s − 1.84·13-s − 0.232·15-s + 0.859·16-s − 1.49·17-s − 0.0727·18-s − 1.25·19-s − 0.382·20-s − 0.0914·22-s + 0.993·23-s − 0.246·24-s − 0.838·25-s + 0.402·26-s − 0.192·27-s − 1.26·29-s + 0.0506·30-s + 1.14·31-s − 0.613·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5322631575$
$L(\frac12)$  $\approx$  $0.5322631575$
$L(\frac{3}{2})$   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 \( 1 + 0.308T + 2T^{2} \)
5 \( 1 - 0.898T + 5T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + 6.64T + 13T^{2} \)
17 \( 1 + 6.17T + 17T^{2} \)
19 \( 1 + 5.46T + 19T^{2} \)
23 \( 1 - 4.76T + 23T^{2} \)
29 \( 1 + 6.78T + 29T^{2} \)
31 \( 1 - 6.37T + 31T^{2} \)
37 \( 1 - 7.16T + 37T^{2} \)
43 \( 1 + 6.00T + 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 4.50T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + 4.95T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 4.36T + 73T^{2} \)
79 \( 1 + 8.63T + 79T^{2} \)
83 \( 1 - 9.05T + 83T^{2} \)
89 \( 1 - 1.29T + 89T^{2} \)
97 \( 1 + 7.81T + 97T^{2} \)
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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

−8.105682629849324701463054829368, −7.35150702302742002915876728139, −6.63943463074799208949864373170, −5.94959587085923772686435766563, −5.01963765668354610024398117914, −4.60158514976495874960350715724, −3.94585403445842199072523515609, −2.61006028509255316812780117068, −1.77693156454107894743491347946, −0.40140645872527802178729847141, 0.40140645872527802178729847141, 1.77693156454107894743491347946, 2.61006028509255316812780117068, 3.94585403445842199072523515609, 4.60158514976495874960350715724, 5.01963765668354610024398117914, 5.94959587085923772686435766563, 6.63943463074799208949864373170, 7.35150702302742002915876728139, 8.105682629849324701463054829368

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