Properties

Label 2-1040-1.1-c1-0-10
Degree $2$
Conductor $1040$
Sign $1$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + 2·3-s − 5-s + 4·7-s + 9-s − 2·11-s − 13-s − 2·15-s + 2·17-s + 6·19-s + 8·21-s + 6·23-s + 25-s − 4·27-s + 2·29-s + 10·31-s − 4·33-s − 4·35-s − 2·37-s − 2·39-s − 6·41-s − 10·43-s − 45-s − 4·47-s + 9·49-s + 4·51-s + 2·53-s + 2·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.516·15-s + 0.485·17-s + 1.37·19-s + 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.371·29-s + 1.79·31-s − 0.696·33-s − 0.676·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 1.52·43-s − 0.149·45-s − 0.583·47-s + 9/7·49-s + 0.560·51-s + 0.274·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.542528445\)
\(L(\frac12)\) \(\approx\) \(2.542528445\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.822167931588977599398615084740, −8.892263963780957263247682851131, −8.050663651453834316178513676641, −7.893612143808235569104846929392, −6.86847438549507433409027285363, −5.26806860730181407315208823154, −4.76761858373058400551305930705, −3.43626038544626492191293151701, −2.65504338827892484384496886087, −1.34491844974665976517057296331, 1.34491844974665976517057296331, 2.65504338827892484384496886087, 3.43626038544626492191293151701, 4.76761858373058400551305930705, 5.26806860730181407315208823154, 6.86847438549507433409027285363, 7.893612143808235569104846929392, 8.050663651453834316178513676641, 8.892263963780957263247682851131, 9.822167931588977599398615084740

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