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  − 2.77·2-s − 1.97·3-s + 5.72·4-s − 3.34·5-s + 5.50·6-s + 7-s − 10.3·8-s + 0.917·9-s + 9.29·10-s − 6.19·11-s − 11.3·12-s − 4.77·13-s − 2.77·14-s + 6.61·15-s + 17.3·16-s + 6.95·17-s − 2.55·18-s + 5.45·19-s − 19.1·20-s − 1.97·21-s + 17.2·22-s − 5.40·23-s + 20.5·24-s + 6.17·25-s + 13.2·26-s + 4.12·27-s + 5.72·28-s + ⋯
L(s)  = 1  − 1.96·2-s − 1.14·3-s + 2.86·4-s − 1.49·5-s + 2.24·6-s + 0.377·7-s − 3.66·8-s + 0.305·9-s + 2.93·10-s − 1.86·11-s − 3.27·12-s − 1.32·13-s − 0.742·14-s + 1.70·15-s + 4.33·16-s + 1.68·17-s − 0.601·18-s + 1.25·19-s − 4.28·20-s − 0.431·21-s + 3.66·22-s − 1.12·23-s + 4.18·24-s + 1.23·25-s + 2.60·26-s + 0.793·27-s + 1.08·28-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  =  $(2,\ 6041,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.09049430945$
$L(\frac12)$  $\approx$  $0.09049430945$
$L(\frac{3}{2})$   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 \( 1 + 2.77T + 2T^{2} \)
3 \( 1 + 1.97T + 3T^{2} \)
5 \( 1 + 3.34T + 5T^{2} \)
11 \( 1 + 6.19T + 11T^{2} \)
13 \( 1 + 4.77T + 13T^{2} \)
17 \( 1 - 6.95T + 17T^{2} \)
19 \( 1 - 5.45T + 19T^{2} \)
23 \( 1 + 5.40T + 23T^{2} \)
29 \( 1 - 1.50T + 29T^{2} \)
31 \( 1 - 6.57T + 31T^{2} \)
37 \( 1 - 2.34T + 37T^{2} \)
41 \( 1 - 3.44T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 4.97T + 47T^{2} \)
53 \( 1 + 0.487T + 53T^{2} \)
59 \( 1 + 1.04T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 7.29T + 67T^{2} \)
71 \( 1 + 0.563T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 2.20T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 - 1.00T + 89T^{2} \)
97 \( 1 + 3.71T + 97T^{2} \)
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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

−7.998652852571154141424825955825, −7.49437246643061938806334295021, −7.27719305906607171907621561518, −6.06006785226279553356836029974, −5.44851756998014903842043888337, −4.70167643949200785590712560954, −3.16036468425357958855859017373, −2.66135020937128645011309752270, −1.20369486300204629341682208876, −0.25412994986459597967863117513, 0.25412994986459597967863117513, 1.20369486300204629341682208876, 2.66135020937128645011309752270, 3.16036468425357958855859017373, 4.70167643949200785590712560954, 5.44851756998014903842043888337, 6.06006785226279553356836029974, 7.27719305906607171907621561518, 7.49437246643061938806334295021, 7.998652852571154141424825955825

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