Properties

Label 2-1040-1.1-c1-0-16
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 + 6·11-s + 13-s + 2·15-s − 6·17-s − 2·19-s + 8·21-s − 6·23-s + 25-s − 4·27-s − 6·29-s − 2·31-s + 12·33-s + 4·35-s + 2·37-s + 2·39-s − 6·41-s − 2·43-s + 45-s + 12·47-s + 9·49-s − 12·51-s + 6·53-s + 6·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s − 0.458·19-s + 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.359·31-s + 2.08·33-s + 0.676·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.304·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s − 1.68·51-s + 0.824·53-s + 0.809·55-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

\(L(1)\) \(\approx\) \(3.000055491\)
\(L(\frac12)\) \(\approx\) \(3.000055491\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−9.625733553856646316066987663990, −8.867824943149867720181872752964, −8.550897617183535641403529782972, −7.60735141810076385697454570168, −6.64130330318779897194146908267, −5.66598464880147773652197285574, −4.35826114760038549855802290715, −3.80767214959855771929683606850, −2.23075150430853632793150689300, −1.64029322900486366162236430580, 1.64029322900486366162236430580, 2.23075150430853632793150689300, 3.80767214959855771929683606850, 4.35826114760038549855802290715, 5.66598464880147773652197285574, 6.64130330318779897194146908267, 7.60735141810076385697454570168, 8.550897617183535641403529782972, 8.867824943149867720181872752964, 9.625733553856646316066987663990

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