Properties

Label 2-1040-1.1-c1-0-18
Degree $2$
Conductor $1040$
Sign $1$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + 3.41·3-s + 5-s + 2·7-s + 8.65·9-s − 4.24·11-s − 13-s + 3.41·15-s − 4.82·17-s + 8.24·19-s + 6.82·21-s − 5.07·23-s + 25-s + 19.3·27-s − 9.65·29-s + 1.41·31-s − 14.4·33-s + 2·35-s − 1.17·37-s − 3.41·39-s − 0.828·41-s + 1.75·43-s + 8.65·45-s − 2·47-s − 3·49-s − 16.4·51-s − 3.17·53-s − 4.24·55-s + ⋯
L(s)  = 1  + 1.97·3-s + 0.447·5-s + 0.755·7-s + 2.88·9-s − 1.27·11-s − 0.277·13-s + 0.881·15-s − 1.17·17-s + 1.89·19-s + 1.49·21-s − 1.05·23-s + 0.200·25-s + 3.71·27-s − 1.79·29-s + 0.254·31-s − 2.52·33-s + 0.338·35-s − 0.192·37-s − 0.546·39-s − 0.129·41-s + 0.267·43-s + 1.29·45-s − 0.291·47-s − 0.428·49-s − 2.30·51-s − 0.435·53-s − 0.572·55-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.477772502\)
\(L(\frac12)\) \(\approx\) \(3.477772502\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 3T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 - 8.24T + 19T^{2} \)
23 \( 1 + 5.07T + 23T^{2} \)
29 \( 1 + 9.65T + 29T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + 1.17T + 37T^{2} \)
41 \( 1 + 0.828T + 41T^{2} \)
43 \( 1 - 1.75T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 3.17T + 53T^{2} \)
59 \( 1 - 5.41T + 59T^{2} \)
61 \( 1 + 7.31T + 61T^{2} \)
67 \( 1 + 0.828T + 67T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 6.82T + 73T^{2} \)
79 \( 1 + 6.82T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−9.662622819620230182144732759631, −9.148036479734114390992896143504, −8.125147413604882015254561565156, −7.76469311471375916617867788197, −6.96898348961327646163272204118, −5.42623865756836789196905133741, −4.52190644736172619743490753065, −3.42415893448044479534166445825, −2.46617969919046215869151829195, −1.72500167905321075740144568774, 1.72500167905321075740144568774, 2.46617969919046215869151829195, 3.42415893448044479534166445825, 4.52190644736172619743490753065, 5.42623865756836789196905133741, 6.96898348961327646163272204118, 7.76469311471375916617867788197, 8.125147413604882015254561565156, 9.148036479734114390992896143504, 9.662622819620230182144732759631

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