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Properties

Label 2-40-40.19-c2-0-6
Degree $2$
Conductor $40$
Sign $1$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
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 + 4·4-s − 5·5-s − 6·7-s + 8·8-s + 9·9-s − 10·10-s − 18·11-s + 6·13-s − 12·14-s + 16·16-s + 18·18-s − 2·19-s − 20·20-s − 36·22-s + 26·23-s + 25·25-s + 12·26-s − 24·28-s + 32·32-s + 30·35-s + 36·36-s + 54·37-s − 4·38-s − 40·40-s − 78·41-s − 72·44-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s − 6/7·7-s + 8-s + 9-s − 10-s − 1.63·11-s + 6/13·13-s − 6/7·14-s + 16-s + 18-s − 0.105·19-s − 20-s − 1.63·22-s + 1.13·23-s + 25-s + 6/13·26-s − 6/7·28-s + 32-s + 6/7·35-s + 36-s + 1.45·37-s − 0.105·38-s − 40-s − 1.90·41-s − 1.63·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{40} (19, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.553034725\)
\(L(\frac12)\) \(\approx\) \(1.553034725\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
5 \( 1 + p T \)
good3 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 + 6 T + p^{2} T^{2} \)
11 \( 1 + 18 T + p^{2} T^{2} \)
13 \( 1 - 6 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 2 T + p^{2} T^{2} \)
23 \( 1 - 26 T + p^{2} T^{2} \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( 1 - 54 T + p^{2} T^{2} \)
41 \( 1 + 78 T + p^{2} T^{2} \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( 1 + 86 T + p^{2} T^{2} \)
53 \( 1 + 74 T + p^{2} T^{2} \)
59 \( 1 - 78 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( 1 - 18 T + p^{2} T^{2} \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−15.77678346778814087418456289815, −15.01878438992949505910846655402, −13.16505296018717364304933665135, −12.80851560461223723565447339392, −11.30955991569730604098580082545, −10.16609465765963409294461457755, −7.921607018737770166253697848351, −6.72879479223605020609989557313, −4.85143091458187152207036527702, −3.25957913652540931152586283582, 3.25957913652540931152586283582, 4.85143091458187152207036527702, 6.72879479223605020609989557313, 7.921607018737770166253697848351, 10.16609465765963409294461457755, 11.30955991569730604098580082545, 12.80851560461223723565447339392, 13.16505296018717364304933665135, 15.01878438992949505910846655402, 15.77678346778814087418456289815

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