Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.61·3-s − 4-s − 0.618·5-s − 2.61·6-s − 7-s + 3·8-s + 3.85·9-s + 0.618·10-s − 2.61·11-s − 2.61·12-s + 5.61·13-s + 14-s − 1.61·15-s − 16-s − 3.23·17-s − 3.85·18-s − 4.09·19-s + 0.618·20-s − 2.61·21-s + 2.61·22-s + 5.09·23-s + 7.85·24-s − 4.61·25-s − 5.61·26-s + 2.23·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.51·3-s − 0.5·4-s − 0.276·5-s − 1.06·6-s − 0.377·7-s + 1.06·8-s + 1.28·9-s + 0.195·10-s − 0.789·11-s − 0.755·12-s + 1.55·13-s + 0.267·14-s − 0.417·15-s − 0.250·16-s − 0.784·17-s − 0.908·18-s − 0.938·19-s + 0.138·20-s − 0.571·21-s + 0.558·22-s + 1.06·23-s + 1.60·24-s − 0.923·25-s − 1.10·26-s + 0.430·27-s + 0.188·28-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  =  1
Selberg data  =  $(2,\ 6041,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 + T \)
863 \( 1 + T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 - 2.61T + 3T^{2} \)
5 \( 1 + 0.618T + 5T^{2} \)
11 \( 1 + 2.61T + 11T^{2} \)
13 \( 1 - 5.61T + 13T^{2} \)
17 \( 1 + 3.23T + 17T^{2} \)
19 \( 1 + 4.09T + 19T^{2} \)
23 \( 1 - 5.09T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 + 2.14T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 - 1.23T + 47T^{2} \)
53 \( 1 + 4.32T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 7.14T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 + 1.85T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 8.76T + 83T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + 4.85T + 97T^{2} \)
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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

−7.966164147079045004430370908122, −7.42825530040303276910612644853, −6.54443210285323835346686876076, −5.60317516235288202814959805631, −4.48049639839874853547067916470, −3.92688796547235443293728004325, −3.21592732367449984372972918221, −2.29510876803931096661565979706, −1.37755701035714148138910374502, 0, 1.37755701035714148138910374502, 2.29510876803931096661565979706, 3.21592732367449984372972918221, 3.92688796547235443293728004325, 4.48049639839874853547067916470, 5.60317516235288202814959805631, 6.54443210285323835346686876076, 7.42825530040303276910612644853, 7.966164147079045004430370908122

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