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  − 2.65·2-s + 2.58·3-s + 5.06·4-s + 1.09·5-s − 6.86·6-s + 7-s − 8.13·8-s + 3.66·9-s − 2.91·10-s − 0.201·11-s + 13.0·12-s + 6.89·13-s − 2.65·14-s + 2.83·15-s + 11.4·16-s − 3.73·17-s − 9.75·18-s − 0.944·19-s + 5.55·20-s + 2.58·21-s + 0.536·22-s + 9.13·23-s − 21.0·24-s − 3.79·25-s − 18.3·26-s + 1.72·27-s + 5.06·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 1.49·3-s + 2.53·4-s + 0.490·5-s − 2.80·6-s + 0.377·7-s − 2.87·8-s + 1.22·9-s − 0.922·10-s − 0.0608·11-s + 3.77·12-s + 1.91·13-s − 0.710·14-s + 0.732·15-s + 2.87·16-s − 0.905·17-s − 2.29·18-s − 0.216·19-s + 1.24·20-s + 0.563·21-s + 0.114·22-s + 1.90·23-s − 4.28·24-s − 0.758·25-s − 3.59·26-s + 0.332·27-s + 0.956·28-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  =  \((2,\ 6041,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.960979157\)
\(L(\frac12)\)  \(\approx\)  \(1.960979157\)
\(L(\frac{3}{2})\)   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 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 - 2.58T + 3T^{2} \)
5 \( 1 - 1.09T + 5T^{2} \)
11 \( 1 + 0.201T + 11T^{2} \)
13 \( 1 - 6.89T + 13T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + 0.944T + 19T^{2} \)
23 \( 1 - 9.13T + 23T^{2} \)
29 \( 1 - 1.11T + 29T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + 0.763T + 37T^{2} \)
41 \( 1 - 2.89T + 41T^{2} \)
43 \( 1 - 2.11T + 43T^{2} \)
47 \( 1 - 1.83T + 47T^{2} \)
53 \( 1 + 4.73T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 5.93T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 7.90T + 73T^{2} \)
79 \( 1 - 1.83T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 4.53T + 89T^{2} \)
97 \( 1 - 3.12T + 97T^{2} \)
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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

−8.411479203058553597735079636609, −7.70039750872057731273226546683, −7.00467646576629383231096509700, −6.39118982456327493954792605538, −5.50089726916242675091813922035, −4.05058463038179126470100356734, −3.22136206864763160686007926968, −2.40414555034135578343324494415, −1.76228821449357595522476436090, −0.956476739963700331957974865385, 0.956476739963700331957974865385, 1.76228821449357595522476436090, 2.40414555034135578343324494415, 3.22136206864763160686007926968, 4.05058463038179126470100356734, 5.50089726916242675091813922035, 6.39118982456327493954792605538, 7.00467646576629383231096509700, 7.70039750872057731273226546683, 8.411479203058553597735079636609

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