Properties

Degree 2
Conductor $ 11 \cdot 17 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.62·2-s − 2.08·3-s + 0.652·4-s − 3.32·5-s + 3.40·6-s − 5.07·7-s + 2.19·8-s + 1.36·9-s + 5.42·10-s + 11-s − 1.36·12-s − 0.291·13-s + 8.26·14-s + 6.95·15-s − 4.87·16-s + 17-s − 2.21·18-s + 7.07·19-s − 2.17·20-s + 10.6·21-s − 1.62·22-s + 1.66·23-s − 4.58·24-s + 6.08·25-s + 0.473·26-s + 3.42·27-s − 3.31·28-s + ⋯
L(s)  = 1  − 1.15·2-s − 1.20·3-s + 0.326·4-s − 1.48·5-s + 1.38·6-s − 1.91·7-s + 0.775·8-s + 0.453·9-s + 1.71·10-s + 0.301·11-s − 0.393·12-s − 0.0807·13-s + 2.20·14-s + 1.79·15-s − 1.21·16-s + 0.242·17-s − 0.522·18-s + 1.62·19-s − 0.485·20-s + 2.31·21-s − 0.347·22-s + 0.347·23-s − 0.935·24-s + 1.21·25-s + 0.0929·26-s + 0.658·27-s − 0.625·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8041\)    =    \(11 \cdot 17 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8041} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8041,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2528118586\)
\(L(\frac12)\)  \(\approx\)  \(0.2528118586\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{11,\;17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad11 \( 1 - T \)
17 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + 1.62T + 2T^{2} \)
3 \( 1 + 2.08T + 3T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
7 \( 1 + 5.07T + 7T^{2} \)
13 \( 1 + 0.291T + 13T^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
23 \( 1 - 1.66T + 23T^{2} \)
29 \( 1 + 4.92T + 29T^{2} \)
31 \( 1 + 5.61T + 31T^{2} \)
37 \( 1 - 8.06T + 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
47 \( 1 - 6.60T + 47T^{2} \)
53 \( 1 - 9.64T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 14.9T + 61T^{2} \)
67 \( 1 - 1.40T + 67T^{2} \)
71 \( 1 - 7.30T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 6.07T + 79T^{2} \)
83 \( 1 + 1.31T + 83T^{2} \)
89 \( 1 + 4.77T + 89T^{2} \)
97 \( 1 - 1.25T + 97T^{2} \)
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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

−7.67730348872054810518513012811, −7.13482035529700651890795550997, −6.83071273017104677371710507361, −5.77636771993089734665404253910, −5.25737499062114033334465327337, −4.07780136181917230219271438251, −3.67367114499311098172568293877, −2.70302409163007033777428989160, −0.929830758415106663716816133651, −0.44484680727674570271303565583, 0.44484680727674570271303565583, 0.929830758415106663716816133651, 2.70302409163007033777428989160, 3.67367114499311098172568293877, 4.07780136181917230219271438251, 5.25737499062114033334465327337, 5.77636771993089734665404253910, 6.83071273017104677371710507361, 7.13482035529700651890795550997, 7.67730348872054810518513012811

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