Properties

Label 2-2535-1.1-c1-0-26
Degree $2$
Conductor $2535$
Sign $1$
Analytic cond. $20.2420$
Root an. cond. $4.49911$
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  + 3-s − 2·4-s − 5-s − 7-s + 9-s + 6·11-s − 2·12-s − 15-s + 4·16-s − 4·19-s + 2·20-s − 21-s − 6·23-s + 25-s + 27-s + 2·28-s − 6·29-s + 5·31-s + 6·33-s + 35-s − 2·36-s + 2·37-s + 11·43-s − 12·44-s − 45-s + 6·47-s + 4·48-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 0.258·15-s + 16-s − 0.917·19-s + 0.447·20-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.11·29-s + 0.898·31-s + 1.04·33-s + 0.169·35-s − 1/3·36-s + 0.328·37-s + 1.67·43-s − 1.80·44-s − 0.149·45-s + 0.875·47-s + 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(20.2420\)
Root analytic conductor: \(4.49911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.558312655\)
\(L(\frac12)\) \(\approx\) \(1.558312655\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−9.008930910832740607560012084607, −8.257123602358388337442955194591, −7.57956281051642326006188534285, −6.55524582534573198925304635730, −5.92304072421045057762692872270, −4.63926054043737451433495067052, −3.95459008578551647729761482809, −3.56366223357914753165489698545, −2.11990586499282594585564805475, −0.78831515756259562175962538725, 0.78831515756259562175962538725, 2.11990586499282594585564805475, 3.56366223357914753165489698545, 3.95459008578551647729761482809, 4.63926054043737451433495067052, 5.92304072421045057762692872270, 6.55524582534573198925304635730, 7.57956281051642326006188534285, 8.257123602358388337442955194591, 9.008930910832740607560012084607

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