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 + 0.381·3-s − 4-s + 1.61·5-s − 0.381·6-s − 7-s + 3·8-s − 2.85·9-s − 1.61·10-s − 0.381·11-s − 0.381·12-s + 3.38·13-s + 14-s + 0.618·15-s − 16-s + 1.23·17-s + 2.85·18-s + 7.09·19-s − 1.61·20-s − 0.381·21-s + 0.381·22-s − 6.09·23-s + 1.14·24-s − 2.38·25-s − 3.38·26-s − 2.23·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.220·3-s − 0.5·4-s + 0.723·5-s − 0.155·6-s − 0.377·7-s + 1.06·8-s − 0.951·9-s − 0.511·10-s − 0.115·11-s − 0.110·12-s + 0.937·13-s + 0.267·14-s + 0.159·15-s − 0.250·16-s + 0.299·17-s + 0.672·18-s + 1.62·19-s − 0.361·20-s − 0.0833·21-s + 0.0814·22-s − 1.26·23-s + 0.233·24-s − 0.476·25-s − 0.663·26-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 \( 1 - 0.381T + 3T^{2} \)
5 \( 1 - 1.61T + 5T^{2} \)
11 \( 1 + 0.381T + 11T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
23 \( 1 + 6.09T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 + 8.85T + 37T^{2} \)
41 \( 1 + 8.94T + 41T^{2} \)
43 \( 1 - 7.70T + 43T^{2} \)
47 \( 1 + 3.23T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 - 0.291T + 61T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 - 4.85T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 1.85T + 97T^{2} \)
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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

−7.903593883955323280920483257250, −7.24082249208048815642605076363, −6.19755158783371892609049136812, −5.61967847425132467550273266263, −5.04506575913172768637878753450, −3.80619497765751653309732269049, −3.29277313806043495190102038987, −2.12493957335690480931053399602, −1.22875976686850573077630225103, 0, 1.22875976686850573077630225103, 2.12493957335690480931053399602, 3.29277313806043495190102038987, 3.80619497765751653309732269049, 5.04506575913172768637878753450, 5.61967847425132467550273266263, 6.19755158783371892609049136812, 7.24082249208048815642605076363, 7.903593883955323280920483257250

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