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Properties

Label 2-1872-52.51-c0-0-1
Degree $2$
Conductor $1872$
Sign $1$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13-s + 2·17-s + 25-s + 2·29-s − 49-s − 2·53-s + 2·61-s − 2·101-s + 2·113-s + ⋯
L(s)  = 1  − 13-s + 2·17-s + 25-s + 2·29-s − 49-s − 2·53-s + 2·61-s − 2·101-s + 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1872} (415, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182017424\)
\(L(\frac12)\) \(\approx\) \(1.182017424\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + T \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )^{2} \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.656955690253039770308071705721, −8.509244514392409736767076060643, −7.907418238542288715463769777006, −7.08906119093741135118384416472, −6.28888456993578558991317303464, −5.26289914734159044212161307119, −4.68037945037740256630262234013, −3.41668235937234969096213356514, −2.64732708754186443775809146998, −1.18285865140344046637603977780, 1.18285865140344046637603977780, 2.64732708754186443775809146998, 3.41668235937234969096213356514, 4.68037945037740256630262234013, 5.26289914734159044212161307119, 6.28888456993578558991317303464, 7.08906119093741135118384416472, 7.907418238542288715463769777006, 8.509244514392409736767076060643, 9.656955690253039770308071705721

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