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  + 2.48·2-s − 3-s + 4.19·4-s − 2.36·5-s − 2.48·6-s + 5.45·8-s + 9-s − 5.88·10-s + 4.46·11-s − 4.19·12-s + 5.01·13-s + 2.36·15-s + 5.18·16-s − 4.88·17-s + 2.48·18-s + 3.64·19-s − 9.91·20-s + 11.1·22-s − 2.22·23-s − 5.45·24-s + 0.599·25-s + 12.4·26-s − 27-s + 2.88·29-s + 5.88·30-s − 1.18·31-s + 1.98·32-s + ⋯
L(s)  = 1  + 1.75·2-s − 0.577·3-s + 2.09·4-s − 1.05·5-s − 1.01·6-s + 1.92·8-s + 0.333·9-s − 1.86·10-s + 1.34·11-s − 1.20·12-s + 1.39·13-s + 0.611·15-s + 1.29·16-s − 1.18·17-s + 0.586·18-s + 0.836·19-s − 2.21·20-s + 2.36·22-s − 0.462·23-s − 1.11·24-s + 0.119·25-s + 2.44·26-s − 0.192·27-s + 0.535·29-s + 1.07·30-s − 0.212·31-s + 0.351·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.742576614$
$L(\frac12)$  $\approx$  $4.742576614$
$L(\frac{3}{2})$   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 \( 1 - 2.48T + 2T^{2} \)
5 \( 1 + 2.36T + 5T^{2} \)
11 \( 1 - 4.46T + 11T^{2} \)
13 \( 1 - 5.01T + 13T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 - 3.64T + 19T^{2} \)
23 \( 1 + 2.22T + 23T^{2} \)
29 \( 1 - 2.88T + 29T^{2} \)
31 \( 1 + 1.18T + 31T^{2} \)
37 \( 1 + 0.748T + 37T^{2} \)
43 \( 1 - 6.98T + 43T^{2} \)
47 \( 1 - 7.64T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 8.70T + 59T^{2} \)
61 \( 1 + 8.35T + 61T^{2} \)
67 \( 1 + 0.0922T + 67T^{2} \)
71 \( 1 - 3.08T + 71T^{2} \)
73 \( 1 - 9.88T + 73T^{2} \)
79 \( 1 - 7.87T + 79T^{2} \)
83 \( 1 - 6.26T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 4.64T + 97T^{2} \)
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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

−7.69942196306514256814277874753, −7.07464025358121963993913626205, −6.30944156689323903044334628908, −6.03550259358055130532047727917, −5.05848274789539829116537935913, −4.23652994559839776249167117199, −3.93209264130612321115673514904, −3.30131491293940674303360225095, −2.08202714801418945009548731635, −0.936592332778388759992096938172, 0.936592332778388759992096938172, 2.08202714801418945009548731635, 3.30131491293940674303360225095, 3.93209264130612321115673514904, 4.23652994559839776249167117199, 5.05848274789539829116537935913, 6.03550259358055130532047727917, 6.30944156689323903044334628908, 7.07464025358121963993913626205, 7.69942196306514256814277874753

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