Properties

Label 2-60e2-4.3-c0-0-2
Degree $2$
Conductor $3600$
Sign $1$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + 2·29-s + 2·41-s + 49-s − 2·61-s + 2·89-s − 2·101-s + 2·109-s + ⋯
L(s)  = 1  + 2·29-s + 2·41-s + 49-s − 2·61-s + 2·89-s − 2·101-s + 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3600} (3151, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.306934344\)
\(L(\frac12)\) \(\approx\) \(1.306934344\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )^{2} \)
97 \( 1 + T^{2} \)
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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

−8.765174895244531193974508901748, −7.964501655175975547360355392106, −7.32128235563975340193434195132, −6.43913369791821294170616881749, −5.84346598347673950005578099283, −4.84522398827095556756981450152, −4.21012682675798903946459521558, −3.15039301255661866883959383893, −2.33208052629464066896006072670, −1.03180259769072802093206375626, 1.03180259769072802093206375626, 2.33208052629464066896006072670, 3.15039301255661866883959383893, 4.21012682675798903946459521558, 4.84522398827095556756981450152, 5.84346598347673950005578099283, 6.43913369791821294170616881749, 7.32128235563975340193434195132, 7.964501655175975547360355392106, 8.765174895244531193974508901748

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