Properties

Degree 2
Conductor $ 7 \cdot 863 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.60·2-s + 0.915·3-s + 4.76·4-s − 4.25·5-s − 2.38·6-s + 7-s − 7.20·8-s − 2.16·9-s + 11.0·10-s + 0.535·11-s + 4.36·12-s + 3.11·13-s − 2.60·14-s − 3.89·15-s + 9.19·16-s + 2.52·17-s + 5.62·18-s + 1.79·19-s − 20.2·20-s + 0.915·21-s − 1.39·22-s + 1.05·23-s − 6.59·24-s + 13.1·25-s − 8.11·26-s − 4.72·27-s + 4.76·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 0.528·3-s + 2.38·4-s − 1.90·5-s − 0.972·6-s + 0.377·7-s − 2.54·8-s − 0.720·9-s + 3.50·10-s + 0.161·11-s + 1.26·12-s + 0.864·13-s − 0.695·14-s − 1.00·15-s + 2.29·16-s + 0.611·17-s + 1.32·18-s + 0.411·19-s − 4.53·20-s + 0.199·21-s − 0.297·22-s + 0.219·23-s − 1.34·24-s + 2.62·25-s − 1.59·26-s − 0.909·27-s + 0.901·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6041\)    =    \(7 \cdot 863\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6041} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6041,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.5778022015\)
\(L(\frac12)\)  \(\approx\)  \(0.5778022015\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;863\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;863\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 - T \)
863 \( 1 + T \)
good2 \( 1 + 2.60T + 2T^{2} \)
3 \( 1 - 0.915T + 3T^{2} \)
5 \( 1 + 4.25T + 5T^{2} \)
11 \( 1 - 0.535T + 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 - 2.52T + 17T^{2} \)
19 \( 1 - 1.79T + 19T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 - 8.42T + 29T^{2} \)
31 \( 1 + 7.53T + 31T^{2} \)
37 \( 1 - 7.37T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 7.82T + 43T^{2} \)
47 \( 1 + 9.30T + 47T^{2} \)
53 \( 1 - 5.88T + 53T^{2} \)
59 \( 1 - 9.36T + 59T^{2} \)
61 \( 1 + 4.82T + 61T^{2} \)
67 \( 1 - 5.73T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + 3.90T + 73T^{2} \)
79 \( 1 - 8.77T + 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 + 2.42T + 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−8.277861020576960237495547454788, −7.69277224559612570868162021812, −7.14699546515511882265651940400, −6.39528572221730966326424556588, −5.32885426773362908397718809653, −4.12982085386409906923242805101, −3.32168531919697802770109088192, −2.74874300807688689733611860770, −1.42369220380611437943570807493, −0.54951212864516851119464419818, 0.54951212864516851119464419818, 1.42369220380611437943570807493, 2.74874300807688689733611860770, 3.32168531919697802770109088192, 4.12982085386409906923242805101, 5.32885426773362908397718809653, 6.39528572221730966326424556588, 7.14699546515511882265651940400, 7.69277224559612570868162021812, 8.277861020576960237495547454788

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