Properties

Label 2-1350-1.1-c1-0-12
Degree $2$
Conductor $1350$
Sign $1$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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-s + 4-s + 2·7-s + 8-s + 3·11-s + 5·13-s + 2·14-s + 16-s − 6·17-s − 4·19-s + 3·22-s + 3·23-s + 5·26-s + 2·28-s + 2·31-s + 32-s − 6·34-s + 11·37-s − 4·38-s − 6·41-s − 4·43-s + 3·44-s + 3·46-s − 3·47-s − 3·49-s + 5·52-s + 12·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.904·11-s + 1.38·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.639·22-s + 0.625·23-s + 0.980·26-s + 0.377·28-s + 0.359·31-s + 0.176·32-s − 1.02·34-s + 1.80·37-s − 0.648·38-s − 0.937·41-s − 0.609·43-s + 0.452·44-s + 0.442·46-s − 0.437·47-s − 3/7·49-s + 0.693·52-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.047734797\)
\(L(\frac12)\) \(\approx\) \(3.047734797\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−9.561875797046048218918886636523, −8.611274002977818360622469998092, −8.151921433043132917691323842011, −6.73490123416920304996637814726, −6.46963573719376188596619565805, −5.33604937618572883955210559321, −4.36987711673591227726964404160, −3.79525861491261319831593936186, −2.42546583038452664737562116099, −1.31423223968587618022895740361, 1.31423223968587618022895740361, 2.42546583038452664737562116099, 3.79525861491261319831593936186, 4.36987711673591227726964404160, 5.33604937618572883955210559321, 6.46963573719376188596619565805, 6.73490123416920304996637814726, 8.151921433043132917691323842011, 8.611274002977818360622469998092, 9.561875797046048218918886636523

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