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  + 1.31·2-s + 3-s − 0.264·4-s − 0.264·5-s + 1.31·6-s − 2.98·8-s + 9-s − 0.348·10-s + 0.401·11-s − 0.264·12-s − 1.84·13-s − 0.264·15-s − 3.40·16-s + 1.89·17-s + 1.31·18-s + 6.30·19-s + 0.0698·20-s + 0.528·22-s − 4.48·23-s − 2.98·24-s − 4.93·25-s − 2.43·26-s + 27-s − 2.74·29-s − 0.348·30-s + 5.89·31-s + 1.48·32-s + ⋯
L(s)  = 1  + 0.931·2-s + 0.577·3-s − 0.132·4-s − 0.118·5-s + 0.537·6-s − 1.05·8-s + 0.333·9-s − 0.110·10-s + 0.121·11-s − 0.0763·12-s − 0.512·13-s − 0.0682·15-s − 0.850·16-s + 0.460·17-s + 0.310·18-s + 1.44·19-s + 0.0156·20-s + 0.112·22-s − 0.934·23-s − 0.608·24-s − 0.986·25-s − 0.476·26-s + 0.192·27-s − 0.510·29-s − 0.0635·30-s + 1.05·31-s + 0.262·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.209882478$
$L(\frac12)$  $\approx$  $3.209882478$
$L(\frac{3}{2})$   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 \( 1 - 1.31T + 2T^{2} \)
5 \( 1 + 0.264T + 5T^{2} \)
11 \( 1 - 0.401T + 11T^{2} \)
13 \( 1 + 1.84T + 13T^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 - 6.30T + 19T^{2} \)
23 \( 1 + 4.48T + 23T^{2} \)
29 \( 1 + 2.74T + 29T^{2} \)
31 \( 1 - 5.89T + 31T^{2} \)
37 \( 1 + 0.497T + 37T^{2} \)
43 \( 1 - 5.86T + 43T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 - 8.75T + 53T^{2} \)
59 \( 1 - 2.50T + 59T^{2} \)
61 \( 1 + 3.86T + 61T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 - 7.37T + 73T^{2} \)
79 \( 1 - 5.34T + 79T^{2} \)
83 \( 1 + 0.106T + 83T^{2} \)
89 \( 1 + 3.79T + 89T^{2} \)
97 \( 1 + 9.72T + 97T^{2} \)
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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

−7.945253490677874374161662493087, −7.46877628106276044142582505705, −6.55625626106617540536860765881, −5.66097074611877612734837244178, −5.26274685436588270753086327861, −4.20502828972309407945557062413, −3.83231112671185332548974236389, −2.95568067188973707092916207759, −2.21157660179966866005681968225, −0.792560062031241744548811686836, 0.792560062031241744548811686836, 2.21157660179966866005681968225, 2.95568067188973707092916207759, 3.83231112671185332548974236389, 4.20502828972309407945557062413, 5.26274685436588270753086327861, 5.66097074611877612734837244178, 6.55625626106617540536860765881, 7.46877628106276044142582505705, 7.945253490677874374161662493087

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