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Properties

Label 2-40749-1.1-c1-0-3
Degree $2$
Conductor $40749$
Sign $1$
Analytic cond. $325.382$
Root an. cond. $18.0383$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 3·5-s + 2·6-s + 3·7-s + 9-s − 6·10-s + 5·11-s − 2·12-s + 2·13-s − 6·14-s − 3·15-s − 4·16-s − 2·18-s − 6·19-s + 6·20-s − 3·21-s − 10·22-s − 9·23-s + 4·25-s − 4·26-s − 27-s + 6·28-s − 29-s + 6·30-s + 2·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 1.34·5-s + 0.816·6-s + 1.13·7-s + 1/3·9-s − 1.89·10-s + 1.50·11-s − 0.577·12-s + 0.554·13-s − 1.60·14-s − 0.774·15-s − 16-s − 0.471·18-s − 1.37·19-s + 1.34·20-s − 0.654·21-s − 2.13·22-s − 1.87·23-s + 4/5·25-s − 0.784·26-s − 0.192·27-s + 1.13·28-s − 0.185·29-s + 1.09·30-s + 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40749 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40749 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40749\)    =    \(3 \cdot 17^{2} \cdot 47\)
Sign: $1$
Analytic conductor: \(325.382\)
Root analytic conductor: \(18.0383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 40749,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.443816528\)
\(L(\frac12)\) \(\approx\) \(1.443816528\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 \)
47 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.75254647494666, −14.22626513010995, −13.69921999951225, −13.45818363151497, −12.38206419527170, −12.09525644056365, −11.32702947287312, −11.00626434277533, −10.46073710862674, −9.973880815934971, −9.495879600157633, −9.010535095625706, −8.377107975244972, −8.118759284034301, −7.276638767700734, −6.576976793983937, −6.222180120193321, −5.768387965915896, −4.828647430618298, −4.359679293354109, −3.657102170688408, −2.246546584301872, −1.789270880210373, −1.494587292498282, −0.5889555865808345, 0.5889555865808345, 1.494587292498282, 1.789270880210373, 2.246546584301872, 3.657102170688408, 4.359679293354109, 4.828647430618298, 5.768387965915896, 6.222180120193321, 6.576976793983937, 7.276638767700734, 8.118759284034301, 8.377107975244972, 9.010535095625706, 9.495879600157633, 9.973880815934971, 10.46073710862674, 11.00626434277533, 11.32702947287312, 12.09525644056365, 12.38206419527170, 13.45818363151497, 13.69921999951225, 14.22626513010995, 14.75254647494666

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