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  + 2.07·2-s + 3-s + 2.31·4-s − 2.30·5-s + 2.07·6-s + 0.652·8-s + 9-s − 4.77·10-s − 2.24·11-s + 2.31·12-s + 4.05·13-s − 2.30·15-s − 3.27·16-s + 1.22·17-s + 2.07·18-s + 4.41·19-s − 5.32·20-s − 4.65·22-s + 6.56·23-s + 0.652·24-s + 0.291·25-s + 8.42·26-s + 27-s + 3.41·29-s − 4.77·30-s − 1.07·31-s − 8.10·32-s + ⋯
L(s)  = 1  + 1.46·2-s + 0.577·3-s + 1.15·4-s − 1.02·5-s + 0.847·6-s + 0.230·8-s + 0.333·9-s − 1.51·10-s − 0.675·11-s + 0.668·12-s + 1.12·13-s − 0.593·15-s − 0.818·16-s + 0.297·17-s + 0.489·18-s + 1.01·19-s − 1.19·20-s − 0.992·22-s + 1.36·23-s + 0.133·24-s + 0.0583·25-s + 1.65·26-s + 0.192·27-s + 0.634·29-s − 0.872·30-s − 0.193·31-s − 1.43·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.747216512$
$L(\frac12)$  $\approx$  $4.747216512$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 2.07T + 2T^{2} \)
5 \( 1 + 2.30T + 5T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 - 4.05T + 13T^{2} \)
17 \( 1 - 1.22T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
29 \( 1 - 3.41T + 29T^{2} \)
31 \( 1 + 1.07T + 31T^{2} \)
37 \( 1 + 1.63T + 37T^{2} \)
43 \( 1 + 4.67T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 - 9.90T + 59T^{2} \)
61 \( 1 - 5.18T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 5.24T + 71T^{2} \)
73 \( 1 - 5.69T + 73T^{2} \)
79 \( 1 - 3.55T + 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 - 5.51T + 89T^{2} \)
97 \( 1 + 3.88T + 97T^{2} \)
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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

−7.967064882856655919644903623367, −7.24535927293442903322178009050, −6.65701570458321410416215636328, −5.66867324082216880767734162525, −5.10630110368576891593301625444, −4.33107301143299464591745424598, −3.55614424485084537046742827374, −3.24764013580982813222844224424, −2.32831049031147362564389318057, −0.903489459514833841238851719565, 0.903489459514833841238851719565, 2.32831049031147362564389318057, 3.24764013580982813222844224424, 3.55614424485084537046742827374, 4.33107301143299464591745424598, 5.10630110368576891593301625444, 5.66867324082216880767734162525, 6.65701570458321410416215636328, 7.24535927293442903322178009050, 7.967064882856655919644903623367

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