Properties

Label 2-8041-1.1-c1-0-89
Degree $2$
Conductor $8041$
Sign $1$
Analytic cond. $64.2077$
Root an. cond. $8.01297$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.40·2-s − 1.47·3-s + 3.78·4-s − 0.987·5-s + 3.54·6-s + 2.06·7-s − 4.29·8-s − 0.827·9-s + 2.37·10-s + 11-s − 5.57·12-s − 2.64·13-s − 4.95·14-s + 1.45·15-s + 2.76·16-s + 17-s + 1.99·18-s + 3.84·19-s − 3.73·20-s − 3.03·21-s − 2.40·22-s − 3.24·23-s + 6.33·24-s − 4.02·25-s + 6.36·26-s + 5.64·27-s + 7.80·28-s + ⋯
L(s)  = 1  − 1.70·2-s − 0.850·3-s + 1.89·4-s − 0.441·5-s + 1.44·6-s + 0.779·7-s − 1.51·8-s − 0.275·9-s + 0.750·10-s + 0.301·11-s − 1.61·12-s − 0.733·13-s − 1.32·14-s + 0.375·15-s + 0.690·16-s + 0.242·17-s + 0.469·18-s + 0.883·19-s − 0.835·20-s − 0.663·21-s − 0.512·22-s − 0.677·23-s + 1.29·24-s − 0.805·25-s + 1.24·26-s + 1.08·27-s + 1.47·28-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

Degree: \(2\)
Conductor: \(8041\)    =    \(11 \cdot 17 \cdot 43\)
Sign: $1$
Analytic conductor: \(64.2077\)
Root analytic conductor: \(8.01297\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8041,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4082714984\)
\(L(\frac12)\) \(\approx\) \(0.4082714984\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
17 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + 2.40T + 2T^{2} \)
3 \( 1 + 1.47T + 3T^{2} \)
5 \( 1 + 0.987T + 5T^{2} \)
7 \( 1 - 2.06T + 7T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
19 \( 1 - 3.84T + 19T^{2} \)
23 \( 1 + 3.24T + 23T^{2} \)
29 \( 1 - 8.79T + 29T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 - 5.60T + 37T^{2} \)
41 \( 1 + 6.40T + 41T^{2} \)
47 \( 1 - 4.14T + 47T^{2} \)
53 \( 1 + 5.18T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 7.38T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 9.93T + 73T^{2} \)
79 \( 1 - 4.82T + 79T^{2} \)
83 \( 1 - 8.60T + 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 - 6.59T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.75986675024691236663910398273, −7.53104494133193822856908730095, −6.61086422796577373364498775481, −6.00397634729185596517381081548, −5.14174418000648198065997991930, −4.46879901866127170656731257680, −3.26334783554716080247294489864, −2.27626189337143221807779476153, −1.36305990535923110158022186759, −0.46304338539659712584564274333, 0.46304338539659712584564274333, 1.36305990535923110158022186759, 2.27626189337143221807779476153, 3.26334783554716080247294489864, 4.46879901866127170656731257680, 5.14174418000648198065997991930, 6.00397634729185596517381081548, 6.61086422796577373364498775481, 7.53104494133193822856908730095, 7.75986675024691236663910398273

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